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^S TRSj 



FLAGS OF ALL COMMERCIAL NATIONS. 




iystr oms P Be ok IE TJPPINCOTT & CO PHILADELPHIA. . 18 61 T. Sinclair's Ml 



^nrkrt-aSook 



OP 



MECHANICS 



AND 



Engineering. 

CONTAINING A MEMORANDUM OF ^P ACTS 
AND CONNECTION %, 0/ 



OF 

Practice and Theorv. 




- BY 
JOHN W) NYSTROM, C. E. 

Eighth JEdition Revised, with additional matter. 

PHILADELPHIA: 
J. B. LIPPINCOTT & CO. 

LONDON : TRUBNER & CO. 

1864. 




Entered according to Act of Congress in the year 1861, by 

JOHN W. NYSTROM, 

in the clerk's Office of the District Court of the United States in and for the 
Eastern District of Pennsylvania. 




i 



PEEFAOE. 

Let every Engineer make his own Pocket-Book as he proceeds 
in study and practice, it will then suit his particular business. 
The present work was compiled in this way continually during 
the Author's professional career. It was originally not intended 
for publication, but had grown so large in manuscript as to be 
inconvenient for the pocket, which circumstance combined with 
repeated requests to publish it, placed it before the public, first 
in the year 1854. Since that time he has travelled in Europe 
for nearly five years, collecting such information for the Pocket- 
Book as to follow up the progress of Engineering profession, and 
no expense has been spared in attaining that object. 

The Author hopes that the introduction of Algebraical formu- 
las instead of written rules will be favourably received. Writ- 
ten rules adopted by English Authors and Engineers are indeed 
excellent ; — but formulas are better, — because they not only tell 
what is to be done, but at a glance, impress the mind with the 
complete operation. If all the formulas in this book were ex- 
plained in words, it would be far too large for the pocket. It is 
avoided to impose upon Engineers to carry elementary expla- 
nations in their pockets which belongs to schoolboys and 
apprentices. 

It is not necessary to understand Algebra for the use of the 
formulas, — only practise the insertion of numerical values, and 
perform the arithmetical operations indicated by the particular 
formula used. The Author has furnished the formulas ready 
to receive what is given, and refund what is required. 

Should there be any letters in formulas not clearly explained 
refer then to the examples. 



(ui) 



ADVERTISEMENT. 



The undersigned is prepared to furnish Drawings and estimates for Pro- 
peller Steamers. He designs Engines and Propellers suitable for any 
desired description of Vessels. Furnishes Drawings of Vessels, with their 
whole internal arrangements distinctly shown in sections and details of any 
desired Scale. 

Several years of experience in this profession enable him to furnish com- 
plete and correct Drawings on very short notice. 

It is very important to have full and complete Drawings before the keel 
of the Vessel is laid, so as to insure unity of action between the Ship and 
Engine Builders, and to afford a clear understanding in reference to con- 
tracts. 

The Drawings will contain all new and useful improvements. 

JOHN W. NYSTROM, 
Civil Engineer, 
Letters will be promptly attended to. Philadelphia. 



Mr. Nystrom, the Author of this Pocket-Book, has been connected with 
me since 1849. as Constructor and Draughtsman, in which capacity 
he has given the greatest satisfaction. During that time my atten- 
tion was constantly called to the readiness and accuracy of his calculations, 
which were made by means of a Pocket-Book then in manuscript. I have 
frequently requested him to publish it, and am now gratified by receiving 
its pages in print. 

In it there Is a Drawing of my Propeller, illustrating the expanding 
pitch first adopted by me, and now generally used. This is the principle 
upon which Propellers have been constructed, under my direction by 
Mr. Nystrom. 

R. F. LOPER. 

March, 1854. 



Contents. 



CONTENTS. 



P\GB 

Accelerated motion - - - 195 

Acceleration, force by - - 205 

Acoustics ----- 235 

Acres ----- 70 

Addition in algebra - - - 13 

Adhesion on railways - - 124 

Adulteration of metals - - 211 

Advertisement - 65 

Aerostatic ----- 230 

Age, moon's - - - - 316 

Air, composition of - - - 290 

Air required for furnaces - 285 

Air pump, dimensions of - - 252 

Algebra ----- 13 

Algebraical signs - - - 12 

Allegation - - - - 20 

Alloys . 183 

Almanac, 19th century - - 311 

Alphabets for headings - - 326 

Anchors and cables - - 116 

Angular velocity - - - 202 

Animal strength - - - 154 

Annuities ----- 68 

Annular double cylinder - 263 

Anomalistic year - - - 310 

Apothecaries' weight - • 71 

Apparent time - - - - 312 

Area and circum. of circles - 104 

Area of plane figures - - 96 

Area of solids - 99 
Arithmetics - - - - 11, 292 

Arithmetical progression - 64 

Astronomy - - - - - 310 

Atmosphere - - - - 230 

Atmospheric columns - - 285 

Atom weight 290 

Audibility of sound - - 235 

Avoirdupois weight - - 71 
Axles and shafts - 175, 181, 182 



Balls, piling of 65, 67 

Balls, weight and capacity of 167 
Bnlloon ----- 233 
Barometer - - - - 231 
Bar iron, weight of - - 187, 189 
Barrel, capacity of - - 71 
Beams, solid and compound - 168 
Beam, walking - - - 93, 103 
Beam, strongest from a log - 174 
Bells, ringing - - - 236, 238 
Belts, leather - - - - 155 
Billiard problem - - - 156 
Binary compounds - - - 291 
Binominal theorem - - 22 

Birmingham wire gauge - - 191 
llast engines - - - - 287 



Blast furnaces - 
Blast, warm and cold - 
Blowing off salt water 
Blowing machines 
Bodies in collision 
Boilers, steam 
Boilers, weight of 
Boiling point - 
Bolts and nuts - - - 
Bramah'a hydraulic press 
Breast wheel, water 
Brick - - - - 
Bushel, standard 



Pack 
. 289 

289 
. 268 

286 

- 156 
280 

- 284 
243 

. 192 
215 

- 224 
170 

- 70 



Cables and anchors - - - 166 

Cables, weight, strength, price 165 

Calculating machine 55 

Caloric - - - - - 240 

Calorimeter in steamboilers - 281 

Capacity, measure of - - 71 

Capacity for Caloric - - 245 

Capacity and weight of balls - 167 

Capacity of solids - 100 
Capacity and weight of materialsl 14 

Cask, capacity of 102 

Cast iron pillars, strength of 163 

Cast iron pipes - - - - 186 

Cast iron, produce of - - 289 

Cast iron girders - - - 173 

Castings, weight of 167 

Catenaria - - - - 145, 149 

Cements, Concretes - 159, 184 

Centre of gravity - 206 

Centre of gyration - - 202 

Centre of oscillation - 200 

Centre of percussion - - 206 

Centrifugal force - - - 198 

Chains, strength, weight, price 165 

Chain, surveying 70 

Chairs for railways - - 170 

Charge of powder - 196 

Charge in blast furnaces - 289 

Chemistry - 290 

Chemical formulas - - 291 

Chemical compounds - - 290 

Chime of bells 233 

'; Chimneys, height of - - - 281 
! Circle, formulas for - - 88, 142 

I Circle, to square a - - 94 

; Circle, area and circumference 104 

1 Circumference of the earth - 293 

! Circular saw - - - - 154 

I Coal, consumption of - - 253 

Coal, weight and bulk of - 282 

j Coefficient of vessels - - 273 



vi 



Contents. 



Page 

Cog' wheels - 178 

Cohesive strength - - - 164 
Coins, American and Foreign 72 
Cold, artificial 244 

Collision of bodies - - - 156 

Colours 69 

Combination - 20 

Combustion of fuel - 283 

Compass, mariners' - 302 

Composition of air, water - 290 
Compounds, chemical - - 290 
Compound interest 68 

Compressive strength - - 162 
Concave and convex mirrors 294 
Concave and convex lenses - 296 
Concretes - 184, 159 

Condenser, fresh water - - 279 
Condensing water - 
Conic sections - 
Conducting power of heat 
Construction of ships 
Consumption of fuel 
Cord of wood - 
Cosecant, natural - 
Cosine, natural - 
Cotangent, natural 
Crank and pin - 
Cube and cuberoot 
Cubic containts - 
Cupola - - - - 
Curvature of the earth 
Curves for railways 
Cut off steam - 
Cut off valve - 
Cycle, lunar - 
Cycle of the sun 



Cycloid 



253 
140 
243 
3iy 
25S 
176 
133 
129 
131 
1S2 
23 
100 
2 So 
303 
116 

- 255 
267 

- 310 
310 

87, 140, 197 



Day and night, length of - - 312 

Decimal fractions to vulgar - 125 

Deaf and dumb alphabet - - 318 

Declination of the sun - - 314 

Degrees of the earth's circle - 70 

Diagram, indicator - - 266 

Diamond ----- 72 

Diameter of wro't iron shafts 175, 181 

Differential calculs 78 

Departure - 300 

Discount or rebate 18 

Displacement of vessels, - 278 

Distances on the American coast 308 

" U. S. railways - - 309 

" in the world - - 306 

" in Europe - - 307 

" of objects at sea - - 303 

" spherical 301 

Distance to the sun - 310 

Division in Algebra - 14 

Dodecahedron 95 

Double cylinder expansion - 264 

Drain, motion of water in • 221 



Dredging Machines 
Dynamical formulas - 
Dynamics 

E 



Paob 

155 

- 153 

152 



Earth, dimension of - - - 298 

Eccentrics - - - - 266 
Economy of expansion of steam 257 

Effects of water, natural - 222 

Ellipse, construction of - 86 

" periphery of - 94 

" formulas for - 142 

Elliptic minor - 294 

Elasticity of beams - 172 

Elevation of external rail - 117 

Embankments - 120 

Engineers' command - - 269 

Engines, different kinds of - 265 

Epact of the year and month 316 

Equation of time - 315 

Equivalents - 290 

Evolute of a circle 87 

Evolution - 22 

Expansion of air - 241 

" liquids 241 

" of solids by heat - 240 

" of steam - - 255 

Exponent of vessels - 273 

Excavation and embankment . 120 

Eyes, long and near sighted . 293 



Faces of the moon - 316 

Falling bodies, table of- - 193 

Fan or ventilator - 288 

Fathom 70 

Feedpump ----- 251 
Fellowship - - - - 18, 19 

Fire engines - 219 

Flags of all nations 1 

Flanges, plate VI. - - - 181 

Floating bodies - 216 

Flour mills ----- 154 

Flooring, beams in - - 169 

Flues and pipes, riveted - - 185 

Flues for steam boilers - - 282 

Fly-wheels ----- 205 

Focal distances of lenses - 296 

" " mirrors - - 294 

Force-pump - - - - 251 

Foot-valves in air pumps - 252 

Forgings by steam hammers 159 

Foreign money 72 
" weights and measures 73 to 76 

Fractions, reduction of - - 125 

Fresh water condenser - - 279 

Friction - - - - - 160 

Fuel, properties of - 282 

Fuel, consumption - - Ji58 

Fulcrum ----- 144 

Funicular machine - - 149 



Contents. 



vil 



Gallon, standard 
Gas, motion of in pipes 
Gearing, force by- 
Gearing, construction of 
Geography - 
Geometry - 
Geometrical progression < 
Girder, compound iron - 

" cast iron - 
Gold metal - 
Golden number - 
Governor - - - - 
Gravitation - 
Gravity, specific 
" centre of 
Gauge, Birmingham 
Gyration, centre of - 

H 

Half-trunk cylinders 

Heat, Caloric - - 

Helix of a screw 

Hexahedron - 

Hemp rope, weight and price 

High water, time of 

Height of the Atmosphere 

Hodgkinson's formula - 

Horse mill - 

Horse power of locomotives • 

" " for work 

u " in machinery • 

M " nominal - 

" " actual - 

Horse, ability of - 
Hose, velocity of water in ■ 
Hydraulic - - - - 
Hydraulic mortar - 
" press - 
" paradox - 
Hydrodynamics - 
Hydrometer - 
Hyperbolical mirror - 
Hyperbola, equation for- 



Paob 

. 70 
235 
147 
178 

. 298 
84 

• 66 
169 i 

. 173 
183 

. 310 
199 

■ 195 
210 
206 
191 
202 



264 

283, 240 

• 93, 94 

95 

165 

316 

- 231 
162 

- 154 
123 

- 152 
262 

- 260 
263 

- 123 
219 

- 218 
159, 184 

- 215 
215 

- 222 
214 

- 294 
143 



Icosahedron- - 95 

Inches and feet, plate I. - - 125 

Inclined plane - 144, 150 
Indicator diagram plate IX. - 266 

Indix of refraction - 295 

Injection water - 253 

Incrustation in boilers - - 269 

Integral calculs 83 

Interest . . - - - 17 68 

Interpolation - 80 

Involution - 21 

Impact of bodies- - - - 157 

Iron or blast furnaces - - 289 



Joints, proportion of riveted - 283 
JonvaPs Turbine 227 



Lap and lead of slide valves - 266 

Latent heat 240 

Lateral strength - - - 168, 171 

Latitude and longitude - - 305 

Law of gravity - 194 

Leap years - - - 311, 314 

Leather belts - 155 

Lenses, optical - 293 

Letters for printing - 318 
Lever ----- 144, 145 

Light ------ 235 

Liquid measure 71 

Locomotive traction - - - 123 

Logarithms - 56 

Logarithm of numbers 58 

" sine, cosine 60 

" tangent, cot. 62 

Longitude into time - - 299 

Log line, length of - - 70 

Lunar Cycle 310 

M 



Magnifying power of lenses - 296 
" Opera-glasses and tel- 
escopes - - 296 
Main slide valve - 266 
Mantissa of logarithms - - 66 
Manual labor - - - - 154 
Mariners' compass - 302 
Maxima and Minima 82 
Meantime - 315 
Measures and weights 70 
Measures, Foreign - - - 73 to 76 
Mechanics - 144 
Mean pressure of steam - - 256 
Meridian altitude, sun's - - 311 
Men's power - - - - 152 
Metacentrum - 216 
Meter, French - 73 
Microscope 293 
Mile, statute and nautical- - 70 
Mills, flour, saw, rolling - 154 

Mill, wind 234 

Mirrors - 293,294 

Momentum in bodies - - - 156 

Momentum, static - 144 
Money ------ 72 

Moon 310 

Moon's age ----- 316 

Moon's faces - 316 

Mortar and Cements - - 159, 184 

Mortar, piece of ordnance - 197 

Motion of bodies - 156 

Multiplication in algebra - 13 

Music 237,239 



Yiii 



Contents. 



N 

Page 

Natural sine and cosine - - 128 

" tangent and cot. - 130 

" secant and cosec. - - 132 

Navigation - 300 

Night and day, length of - - 312 

Nominal horse power - - 260 

Notation of numbers - - 11 

Nuts and bolts 192 

Ny Strom's Calculator- - 65 



Octahedron - 95 

Opera glassas - 295,297 

Optics 293 

Ordinates for Railway curves - 115 
Oscillation of Pendulum - 201 
Overshot wheel - - - - 225 



It, value of, to 127 decimals - 88 

Paddle wheels - - - - 278 

Paper, drawing and tracing - 69 

Parabola, to construct a - 87, 143 

Parabolic ships - - 319 

Parabolic mirror - 294 

" vein 221 

Paradox, hydraulic - 215 

Pattern makers' rule - • 167 

" for castings - 167 

Peal of eight bells 238 

Pendulum - - - - - 200 

Percussion, centre of 206 

Periphery of an ellipse 94 

Permutation - 19 

Phoenix Iron Co., beams - - 168 

" " Rolled iron 170 

Piling of balls and shells - 65, 67 

Pile-driving - - - - 158 

Pipes, motion water in - 218, 228 

" motion of gas in - - 235 

" cast iron, weight of - 186 

" and flues, riveted - - 185 

" steam - 254 

Pitch of propellers - 271 

" of screws 93 

"■ of spirals - - - 87, 94 

" of teeth in gearing - - 178 

Planetary system - 313 

Plane sailing - 300 

Plane inclined - - - - 150 

Points on the compass - - 302 

Polygons - - - - - 103 

Polyhedrons - ... 95 

Poncelet's water wheel - - 224 

Population of the Earth - - 298 

" of Countries - 304 

Ports, steam - - - - 254 

Powder, charge of - - - 196 

" composition of - - 290 

11 forced by - - 196 



Page 

Power of water - 222 

" of engines - - 260, 263 

" nominal horse - 260 

" actual horse - 263 

" of locomotives - - 123 

" of men 152 

" of ahorse - - - 123 

11 in moving bodies - - 196 

" definition of - 152 

M for steamboats - - 276 

" for different mills - - 154 

" for punching - - 285 

" magnifying ■■-•■-• 296 

" reflecting of heat - - 243 

Pound, avoirdupoise, troy- - 71 

Press, hydraulic - 216 

Printing letters for headings - 318 

Progression, Arithmetical - 64 

" Geometrical 66 

Price of materials - 165, 169, 170, 282 

Propellers, screw - 270 

Proportion - 15, 16 

Provision ----- 77 

Pulleys 148 

Pump, air, and feed - - 252 

Punching iron plates - - - 285 

Q 

Quantity, definition of - * 11 

R 

Radiating power of heat - - 243 

Radius of the Earth - - 298 

Roads, traction on 122 

Rail, elevation of the outer - 117 
Railroads - - - 115 to 124 

Rails, spring of 117 

Railway curves - - - - 116 

Rails, weight and price of - 170 

Rails for streets - - - - 170 

Ram in pile driving - - 158 

Range of a cannon ball - - 197 

Rebate or discount - - - 18 

Rectangular beams from a log - 174 

Reduce inches to feet - - 125 

Reflecting power of heat - - 243 

Refractive indices - - - 295 

Resistance in Railway curves - 124 

Resultant of forces - 147 

Retarded motion- - - - 195 

Ringing bells - 236 

Riveted joints, proportion of - 283 

Roebling's wire ropes - - 165 
Rolled iron, weight, size,price 170, 184 

Rolling mills 154 

Roman cement - - - - 159 

Roman notation - - - 12 

Ropes, strengtlL, weight, price - 165 

Roots, square and cube 23 





Contents. 


ix 


s 


Page 




Paob 


Safety valve - 


- 254 


Subtraction in algebra - 


13 


Sailing distances 


306, 398 


Sun, set and rise - 


312 


Salt water in boilers - 


- 268 


" cycle of the - 


310 


Sashbars, iron, for windows 


170 


" distance to - 


310 


Saturation in boilers - 


- 269 


Sun's altitude - 


311 


Saw, circular and alternative 154 


" declination - 


314 


Saw mill water wheel - 


225 


Superheating steam 


259 


Screw jack - 


- 151 


" apparatus - 


259 


" propeller 


270 


Surface condenser - 


279 


" helix - - - 


93, 94 


Surveying chain - 


70 


" force by - 


145, 151 


Surface of solids - 


99 


11 proportions of - 


- 192 






Secant, natural 


132 


T 




Sections, conic - 


- 140 






Segments of a circle 


110 


Talon, to construct a - 


84 


Shafts, strength of 


- 181 


Tangent, natural - 


130 


" diameter of - 


175 


" logarithm - 


62 


Sheering iron plates - 


- 285 


Teeth for gearing - 


178 


Ships construction of 


319 


Telescope, astronomical - 


297 


Shrinkage of castings - 


- 167 


Temperature on the Earth - 


242 


Sideral year - 


310 


" of substances 


243 


Signs, Algebraical 


- 12 


Tempering steel - 


183 


Simple interest 


17 


Tetrahedron - 


95 


Simple substances 


- 290 


Thermometer - 


242 


Sine and cosine, natural 


128 


Threads, number of per inch - 


192 


" " logarithm 


- 60 


Thrashing Machine - 


154 


Slide valves - 


266 


Time of high water - 


316 


Slip of propellers 


- 273 


" apparent - 


312 


Slope of embankments - 


120 


" equation of 


315 


Smelting point - 


- 243 


" difference in - 


299 


Solders, for bracing 


183 


Tonnage of vessels - 


278 


Soldiers' steps - 


- 152 


Tracing paper - 


69 


Solids, capacity of - 


100 


Traction on roads - 


122 


Sound, acoustic - 


- 235 


Traveling distances - 307 


, 309 


Soundings to low water - 


317 


Triangles, formulas for 


134 


Specific gravity - 210 


212, 290 


" different kinds 


85 


Specific caloric 


244 


11 by diagrams 


136 


Speed & horse power of steamers 276 


Trigonometrical functions - 


127 


Spherical distances - 


301 


Trigonometry, plane - 


125 


M mirrors 


- 294 


" spherical - 


136 


" trigonometry - 
Spiral, construction of 


136 


Tropical year - 


310 


87, 94 


Troy weight - 


71 


Spring of rails - - - 


117 


Tubes, iron, lap welded 


282 


Square a circle, to 


- 94 


Turbines, Jonval's - 


226 


Square and square root - 


23 


Tuyeres in furnaces « 


286 


Stability of floating bodies 


- 216 






Static momentum - 


144 


U 




Steam - 


- 246 






" boilers - 


280 


U. S. standard weight, measure 


70 


" condenser - 


- 279 


Undershot water wheel 


224 


" engines - 


265 






" expansion of - 


- 255 


V 




" hammers 


159 






" pipes and ports 


- 254 


Valves, air pump - • - 


252 


" superheated - 


259 


u safety - 


254 


" ship performance - 


- 274 


" slide - 


266 


" table - 


248 


" blast engines 


286 


Steel, tempering of - 


- 183 


Vein of water, contracted 


221 


Strength of materials - 162 to 176 


" " parabolic - 


221 


Strength of animals - 


- 154 


Ventilator, fan - 


288 


Stubends, plate V. - 


181 


Vessels, resistance to 


274 


Stuffing boxes, plate VI. - 
i 


- 181 


" tonnage of 


278 



Contents. 



Page 

Velocity, angular - 202 

" of gas in pipes - 235 
" of water in pipes - 218, 228 

" of men- - - - 152 

Vibration of pendulum - - 201 

" in music - 237 

Volume expansion - 241 

Vulgar fractions to decimals - 125 



W 



93, 103 



Walking beam - 

Water colours - 69 

" composition of 290 

" fresh, condenser - - 279 

" power of 222 

" motion of in pipes - 218, 228 

" wheels - 222, 225 

Wedge, force by - - - 145, 151 

Weight and measure 70 
" and capacity of balls - 167 
" of round, square iron 184 

" wrought iron pipes - 185 

" cast iron pipes - - 186 

" cast iron cylinders - 185 



P>GS 

Weight flat rolled iron - - 187 

" copper bolts - 190 

" per square foot - 191 

" and capacity - - 114, 210 

" of castings - 167 

" oi materials - 1S1 to 184 

" of steam hammers - 159 

" of engines and boilers - 284 

Weirs, water flowing over - 222 

Western steam boilers - - 283 

Wheels, paddle 278 

" water - - - - 224 

Wind, force by- 233 

" mills 234 

" velocity of - - - 232 

Window sashes, iron - 170 

Wire gauge - - - - 191 

Wire ropes, strength, price - 165 

Wood, cord of - - - - 176 

Wrought iron beams - 168 



Yard, feet and inches - 
Years, different kinds of 



70 
310 



Mathematics. H 



INTRODUCTION. 

Quantity is that which can be increased or diminished by augments or 
abatements of homogeneous parts. Quantities are of two essential kinds, 
Geometrical and Physical. 

1st, Geometrical quantities are those -which occupy space ; as lines, surfaces, 
solids, liquids, gases, &c. 

2nd, Physical quantities are those which exist in the time but occupy no space, 
they are known by their character and action upon geometrical quantities ; as 
attraction, liglit, heat, electricity and magnetism, colours, force, power, &c, &c. 

To obtain the magnitude of a quantity we compare it with a part of the same, 
this part is imprinted in our mind as a unit, by which the whole is measured 
and conceived. No quantity can be measured by a quantity of another kind, 
but any quantity can be compared with any other quantity, and by such com- 
parison arises what we call calculation or Mathematics. 



MATHEMATICS. 

Mathematics is a science by which the comparative value of quantities 

are investigated ; it is divided into : 

1st, Arithmetic* — that branch of Mathematics, which treats of the nature 
and property of numbers ; it is subdivided into Addition, Subtraction, Multiplicar 
tion, Division, Involution, Evolution and Logarithms. 

2nd, Algrehra, — that branch of Mathematics which employs letters to repre- 
sent quantities, and by that means performs solutions without knowing or 
noticing the value of the quantities. The subdivisions of Algebra are the same 
as in Arithmetic. 

3rd, Geometry) — that branch of Mathematics which investigates the rela- 
tive property of quantities that occupies space ; its subdivisions are Zongemetry, 
Planemetry, Stereometry, Trigonometry, and Conic Sections. 

4th, Diffcreiitial-calculsj — that branch of Mathematics, which ascer- 
tains the mean effect, produced by group of continued variable causes. 

5th, Integral-calculs$ — the contrary of Differential, or that branch of 
Mathematics which investigates the nature of a continued variable cause, that 
has produced a known effect. 



ARITHMETIC. 

Tlie art of maneuvering numbers, and to investigate the relationship of 
quantities. 

Figures — 1, 2, 3, 4, 5, 6, 7, 8, 9. Arabic dignets, about nine hundred years old. 

Ciphers — 0, 0, 0. Sometimes called noughts, it is the beginning of figures and 
things. 

Number is the expression of one or more figures and ciphers. 

Integer is a whole number or unit. 

Fraction is a part of a number or unit. 

When figures are joined together in a number, the relative dignity expressed 
ty each figure, depends upon its position to the others. Thus, 



1 •§ I s ! 

674,385 ; 496,345 ; 695,216 ; 505,310 : 685, 3 67 ; 



12 



Notation. 



Notation is the setting down of any number by figures and ciphers. 

Numeration is the reading of any number in words, from the expression of 
figures and ciphers. 

Characters which describe tJie operation by numbers {significations). 
= Equality, as 6=6, reads 6 is equal to 6. 

-J- Plus, Addition, 3 +6=9 

— Minus, Subtraction, 6 — 2=4 

X Multiplication, 3X^=12 

-f- or : Division, - - - • - - - -15: 5=3 

sj Square root, »J 9=3 

}/ Cube root, */ 8=2 

> Greater, - - - - - - - - - - 8 >• 4 

< Less, 6<9 



ROMAN NOTATION. 

The Romans expressed their numbers by various repetitions and combin- 
ations of seven letters in the alphabet ; as, 
1=1. 



2=11. 

3=ni. 

4=IV. 

5=V. 

6=YI. 

7= VII. 

8=VIII. 

9=IX 
10=X. 
20=XX. 
30=XXX. 
40=XL. 
50=L. 
60=LX. 
70=LXX. 
80=LXXX. 
90=XC. 
100=C. 

500=D, or L0. 



1,000= 

2,000= 

5,000= 

6,000= 

10.000= 

50,000: 

60.000: 

100,000: 

1,000,000= 
2,000,000 



=M, or CD. 

=MM,orIIOOO. 
=V 1 orLDO. 
= VI, or MMM. 
=X, or COO. 
=L. or LOOO- 
=LX,orMMMD. 
=C, or COOO. 
=M. or COOOO. 
-MM, or MMOOO. 



f As often as a character is repeated, 
(.so many times is its value repeated. 



( A less character before a greater 
< diminishes its value, as IV=I from 
(V, or 1 substracted from 5=4. 



( A less character after a greater in- 
creases its value, as XI=X+I, or 
(10+1=11. 



f For every O annexed, this becomes 
( 10 times as many. 



A bar, thus, — over any number, increases 
' it 1000 times. 



Examples.— 1854, MDCCCLIV. 

524,365, DXXIYCCCLXV. 
An imperfection in the Roman Notation consists in that, there is no significa- 
tion for the cipher as in the Arabic notation. 



Algebra. 13 



ALGEBRA. 



In Algebra -we employ certain characters or letters to represent quantities. 
These characters are separated by signs, which describe the operations ; and by 
that means, simplify the solution. 

1. Whatever the value of any quantity may be, it can be represented by a 
character, as a. Another quantity of the same kind, but of different value, be- 
ing represented by 6. The sum of these two quantities is of the same kind but 
of different value. 

For Addition we have the algebraical sign -f , (plus) which, when placed 
between quantities, denotes they shall be added; as a+6, reads in the 
algebraical language, " a plus b" or a is to be added to b. 

Another algebraical sign =, (Equal) denotes that quantities which are placed 
on each side of this sign, are equal. Let the sum of a and b be denoted by the 
letter c; then we have, 

a+b=c. 

This composition is called an algebraical equation. The quantity on each side 
of the equal sign is called a member, as a+b, is one member, and c, the other. When 
one of the members contains only one quantity, that member is generally 
placed on the first side of the equal sign, and its value commonly unknown ; 
but the value of the quantities in the other member being given, as a=4, and 
6=5, then the practical mode, to insert numerical values in algebraical equa- 
tions, will appear; as, 

Equation, c=a+b, 

4+5=9, the value of e. 

2. The sum of three quantities a; b, and c, is equal to d, then 

Equation, d=a-\-b-\-c, 

4+5+9=18, the value of d. 

3. For Subtraction we have the algebraical sign, — , (minus) which, when 
placed before a quantity, denotes it is to be subtracted as, a — b, reads in the 
algebraical language " a minus &," or from a, subtract b. Let the difference bo 
denoted by the letter c ; and a=8. 6=3 

Equation, c=a — b, 

8 — 3=5, the value of c. 

4. From the sum of a and b, subtract c, and the result will be d ; then, 

Equation, d=a+b — c, 

8+3—5=6, the value of d. 

5. When two equal quantities are to be added, as a+a, it is the same as to 
take one of them twice, and is marked thus 2a. The number 2 is called the 
coefficient of the quantity a. If there are more than two equal quantities to be 
added, the coefficient denotes how many there are of them; as, 

Equation, - - - - a+a=2<z, 
" a+a+a=3a, 

" a+a-j-a-\-a=4a, 

dc., (fa. 

When the quantities are separated by the signs, plus, or minus, they are 
called terms. 

6. Multiplication. — When a quantity a, is to be multiplied by another 
quantity b, then a and b are called factors ; and separated by no sign as ab ; 
which denotes that a is to be multiplied by 6 ; but when the values of a and b 
are expressed by numbers, they are separated by the sign X (Multiplication) ; the 
result from Multiplication is called the product. Let a=8, and 5=6, and the pro- 
duct of a and b, to be c, then, 

Equation, c—ab, 

8X6=48, the value of c. 

7. The product of a and b. is to be multiplied by c, and the latter product will 
bo equal to d ; then, 

Equation, d=abc, 

8X6X48=2304, the value of d. 



14 Algebra. 

8. The sum of a and b, is to be multiplied by c, and the product will be d; 
then, 

Equation, d = c (a+b), 

48 (8+6) = 672 the value of d. 

When the sum of two or more quantities is to be multiplied by another quan- 
tity, the sum is to be enclosed in parentheses, and denotes itself to be one factor. 
The other factor is to be placed on the outside of the parentheses, as seen in the 
preceding example. 

9. To the product of a and c, add b, and the result will be d; then, 

Equation, d= ac +6, 

8X^8+6 = 390 the value of d. 

Be particular to distinguish the two Examples 8, and 9. 

10. The sum of a and b, to be multiplied by the sum of a and c ; the product 
will be d; then, 

Equation, d = (a +6) (a + c), 

(8+6) (8+48) = 784. 

11. The sum of c and b, to be multiplied by the difference of c and a; the re- 
sult will be d ; then, 

Equation, d = (c+b) (c — a), 

(48+6) (48— 8) = 2160. 

12. Division. — When a quantity a, is to be separated into b equal parts, the 
numbers of parts or b, is called the divisor, and the value of each part, is called 
the quotient. The sum of the parts or the whole quantity a, is called the dividend ; 
a and b, is separated by the sign : (Division) ; as a : b, reads in the algebraical 
language, "a divided by 6." Let the quotient be denoted by the letter c; and 
a=18, 6=6, then, 

Equation, c — a:b, 

18 : 6 = 3 the quotient c. 

In Algebra it is found [more convenient to set up Division as a fraction, then 
it will appear as, 

13. Divide a, by c, and the quotient will be b. Then, 

c 
18 
•5* =s 6 the quotient 6. 

14. The product of a and b, to be divided by c ; and the product will be d. 
Then, 

ah 
Equation, d = » 

18X6 
— = 36. 

15. The sum of d and b, to be multiplied by c, and the product divided by a; 
then the result will be e. 

c(d+b) 
Equation, e = — -= , 

3 (36+6) 

18 ~ 7, 

16. From the product of a and c, subtract 36 ; divide the remainder by the 
difference of a, and c; the result will be h. 



Proportion. 15 



ac—Sb 
Equation, h — 



a — c 
18X3—3X6 



= 2.4. 



18-3 

An old man said to a smart boy, " How old are you?" to which he replied.— 
" To seven times my father's age add yours, divide the sum by double the 
difference of yours and his, and the result will be my age." 

Letters will denote, 
a = the old man's age, 
6 = the father's age, 
c = the boy's age. Then, 

„ ,. 75-fa ._ _ , 

EqwiXvmy c — ■— - — rr the boy's age. 

A [Q, — 0) 

Now for any number of years of the old man and the father, will be a corres- 
ponding age of the boy ; suppose, 

a = 73 years the age of the old man, 
b = 57 years the father's age. 
Require the boy's age. 

7X57+73 ,-,_ 
C =2-(73=57)= 14 * yearS - 



PROPORTION. 



The relative value of two quantities, is obtained by dividing one into the other, 
and the quotient is called the ratio of their relationship. If the ratio of two 
quantities is equal to the ratio of two other quantities, they are said to be in the 
same proportion ; as, 

a : b = c : d , 
reads in the algebraical language " a is to b as c is to d." — a, 5, c, and d, are call- 
ed terms, of which a is the first, b the second, c the third, and d the fourth term. 
The first and fourth are called " the outer terms," and the second and third, 
"the inner terms." The whole is called an "analogy." 

A property in the nature of analogies is, that the product of the outer terms 
ad, is equal to the product of the inner be Suppose a = 4,& = 9, c=12, 
d — 27. 

4:9=12 :27, 
ad=bc, 4X27=9X12. 

If any one of the four quantities are unknown, its value can be calculated 
by the other three ; as, 

d 27 



& = 




= 12. 

= 27. 



16 



Proportions. 



To Alternate a Proportion. 
If a: b = c : d, 
then a : c = b : d, 
and ad = be. 



To Inverse a Proportion. 
If a : b = c : d 7 
then b : a = d : c, 
and Z>c = «</. 



To Multiply a Proportion. 
If a: b= c: d, 

then 7i# : ?20 = wc : nd, 

, a £ c d 
and —;-= — :_. 
n 7i n n 



Compared Proportions. 
If a : b = c : d 9 

and c : d = e :f 
then : b = e :f. 



To Reduce a Proportion. 
If a: b = c: d, 

then na : nb = mc : md, 

.abed 

and — ; = — : _. 

n n mm 



Continued Proportion. 
If a : b = c : d = e : f f 
then of = 6e = cd, 
and ad =£c, cf= de. 



To Compound Proportions. 

If a : d -= c : d y 
and e:f=g:h, 
then ae : <//*= eg- : dA. 



To Compare Proportions. 
If a: b= c: d 7 
and e:f=g:hy 

then -:-«—:- 

* / g h 



To Combine a Proportion. 

If a: b = c: dy 

then (a+6) : b = (c+d) : d y 
and d {a+b) = b (c+d). 



To Dissolve a Proportion. 

If a: b=~ c: dy 

then (a— b) : o = (c— d) : d, 
and d (a— b) = £ (c— d). 



To Combine a Proportion 

Inversely. 
If a: b = c: dy 
then a : b -. (a+c) : (o-f d), 
and a (6+d = b (a+c). 



To Dissolve a Proportion 

Inversely. 
If a: b = c: d y 
then a: b = (a—c) : {b—d) f 
and a (o— d) = 6 (a—c). 



If 



a : Z> = d : c 9 



then a: &-=- :f 

c a, 

and — : \= c: d. 



and 



AC = bd. 



To Find the mean Proportion, 
a : x = x : b 
then x = y/~ab ' 

x is the mean Proportion. 



Proportion of Square and Square 
Root. 
If a: b= c: dy 

then a 2 :P = c: d l 
and sfa~\ sfb~= <f~c\J~d. 



Propoi Hon of any Power or 

Root. 
If a : b = c : dy 

then a n : b n = c 11 :_d a , 
and^T: vT"= ^ti.^/1. 



Simple Interest. 



SIMPLE INTEREST. 

Interest is a profit on money which is lent for a certain time. 
Letters will denote, 
c = the standing capital, or lent money. 
r = interest on the capital c, 
p = per cent, on 100 in the certain time. 

Analogy, c : r = 100 : p. 

If j) is the per cent, on 100, in one year, then t = time in years for the stand- 
ing capital c, and the interest r. 

Analogy, c : r = 100 : pL 

"From this analogy we obtain the equations, 

Interest, r = -^p - - • - 1, 

100 r 
Per cent., p = — — — , 2, 

Capital, c = , • • • • -3, 

pt 

m- . 10° r 

Time in years, t = , • •-•*. 

cp 

Now for any question in Simple Interest, there is one equation which gives the 
answer. If the time is given in months, weeks, or days, multiply the 100 cor- 
respondingly by 12, 52, 365. 

Example 1. What is the interest on $3789.35, for 3 years and five months, at 
6 per cent, per annum? 

t = 3X12+5 = 41 months, from the Equation 1, we have, 

7nW , r = 3789.35X6X4 U 7 . 681I)onargj 

Example 2. A capital c = $469.78, gave an interest r = 150.72 dollars, in a 
time t = 4 years and 7 months. Require the per centage per annum? 
t = 4X12+7 = 55 months, from Equation 2, we have, 

t> , 12X100X150.72 _ 

Per cent, p = 469-78 x 55 - 7 per cent. 

Example 3. What capital is required to give an interest r = 345 Dollars in 6 
years, at 5 per cent, per annum ? From the Equation 3^ we have, 

Capital, c = 10 ^ 45 = $1150. 

Example 4. A capital c = $2365 shall stand until the interest will be r = 650 
Dollars, at p = 6 per cent, per annum. How long must the capital stand? 
From the Equation 4, we have, 

m- 100X550 

Tl ™> ^ 236^X6 ==3 ' 876yearS * 

12X0.876 = 10.512 months, 4X0.512 = 2.048 weeks, the time t = 3 years, 10 
months, and 2 weeks. 

— - _ . 



Rebate or I/iscount.— Fellowship. 



REBATE OR DISCOUNT. 

Rebate or Discount is an allowance on money, which is paid before due. 
a = amount of money to be paid in the time t. By agreement the amount is 
paid with a capital c, at the beginning of the time t, but discounted a Rebate r, at 
p per cent, so that the interest on the capital c, &tp per cent., should be equal to 
a = c+r. 
apt 

r = m^t • • • - s, 



the Rebate r, in the time t. 
Rebate, 



Capital, 
rer cent., 
Time, 
Amount, 
Amount, 



100 a 

^lOO+^tf 

= 100 (a—c) 



. 100 (a— <:) 
< = — cp—>' 



100 



(ioo+i>0, 



a - — (100+1»0, 



7, 
8, 

10. 



Now for any question in Rebate or Discount, there is one equation that will 
give the answer. 

Example 5. A sum of money, a = 78460 dollars is to be paid after 3 years and 
6 months, but by agreement payment is to be made at the present time. What 
will be the Rebate at 7 per cent. 

t> r , 78460X7X3.5 MBMmkM 

Rebate, r - loo ; 7x3>5 = $15439.91. 



FELLOWSHIP. 

Fellowship* or partnership, is a rule by which companies ascertain each 
fellow's profit or loss, by their stock. Each fellow's part in the stock is called 
his share. The sum of shares is called the stock. 
Fellowships are of two kinds, Simple and Double. 

Simple Fellowship, when there is no regard to the time, the shares or 
stock is employed. 

Letters will denote, 
A = share of either one fellow. 
a = profit or loss on the share A. 
S = stock or the sum of the shares. 
s = gain or loss on the stock & 
Then A:a = S:t. 

aS 



Share, 
Profit or loss, 
Stock, 
Gain or loss, 



A=- 



s 



- 11, 



° = 1T' 12 > 

S=*— , 13, 

a 



Permutation. 



19 



Example 1. A person had invested A = $11635, in a stock S = $64800, 
which gave a gain of s = 13S64. "What will he the profit of the person's 
share? 



Profit, 



a-2*ggZ-pm» 



Double Fellowship. When the different shares are employe! at a dif- 
ferent length of time, eacn share is" multiplied hy its time employed, and the 
product is the effect of the share. 

Letters will denote, 

t = time for the employed share A. 

T = meantime for the employed stock S. 

e = effect of the share A. 

a = profit of the effect e. 

E = effect of the stock. 

s = gain of the effect E. Then, 



Effect of A y 
Profit of e, 
Effect of S, 
Gain of E, 



aE 



E=- 



aE 



t:a = 


• E:s. 




15, 


Time, 


aE 
As, 


16, 


Share, 


aE 

A=s ir 


IT, 


Meantime, 


es 


18, 


Stock, 


„ es 



19, 
. 20, 

21, 
. 22. 



Example 2. A canal is to he dug, and requires an effect E = 76850 (men and 
days) to he accomplished ; after that it will give a gain 5 = 12390 Dollars. An 
employer has A = 168 laborers. How many days must those laborers he em- 
ployed at the canal, that the employer will obtain a profit a = $5000 ? 



Time, 



5000X76850 
168X12390 



= 184.6 days. 



PERMUTATION. 

Permutation is to arrange a number of things in every possible position. 
It is commonly used in games. 

Example 1. How many different values can be written by the three ciphers 

1X2X3 = 6 different values, namely, 

123, 132, 213, 231, 312, 321. 
With any three different ciphers can be written six different values. Any 
three things can be placed in 6 different positions. 

Example 2. How many names can be written by the three syllables mo, ta, 
la ? The answer is, — Motala, Molata, Tamola, Talamo, Lamota, Latamo. 

Example 3. How many words can be written by the five syllables, mul tip, It, 
ca, turn t 

1X2X3X4X5 = 120 words, the answer. 



10 Combination.— Alligation. 



n — p The accompanying table shows the 

1 = 1 permutation of different numbers of 

2 = 2 things up to 14 ; which will be con- 
S = 6 venient in the next coming examples 

4 = 24 in combination 

5 = 120 

6 = 720 

7 = 5040 

8 = 40320 

9 = 362880 

10 = 3628800 

11 = 39916800 
#12 = 479001600 

13 = 6227020800 

14 = 87178291200. 



COMBINATION. 

Combination is to arrange a less number of things out of a greater, in 
fc*©jry possible position. It is commonly used in games. 

Examph 1. How many different numbers can be set up by the nine ciphers, 
1, 2, S, 4, 5, 6, 7, 8, 9, and three ciphers in each number? 

9X8X7 

-=— - — - = 84 different numbers. 
1X^X«* 

Example 2. How many different Tariations can a player obtain his cards, when 
the set contains 52 cards, of which he receives 8 at a time ? 

52X51X S0X 40X48X47X46X45 7 , 9 „ ft1 _ ™ r ; n+1 -™ c 

-1X-2XTSX-4X 5X 6 X 7 X 8 = ' 5253815 ° ™™ tlons ' 

If they are four players, and jpr. 4 = 24, they can play 24X752538150 

=1S,060;915,600 different plays. 

If it takes half an hour foi each play, and they play 8 hours per day, it will take 

18060915600 

— - — = 1125807225 days = 3;092,622 years. 

2X8 



ALLIGATION. 

Alligation is to mix together a number of different things of different 
price or value, and ascertain the mean value of the mixture ; or from a given 
mean value of a mixture ascertain the proportion and value of each ingre- 
dient. 

Let the different things be a, b } c, and d, &c, their respective price or value 
per unit, z, y, x, and w, &c. 
A — a+fc+c-f d &c, the sum of the things. 
P = mean value or price per unit of A. Then, 

AP~az-\-by-\-cx-\-dw-\-<&c, ----- 1, 
and 

az-\-by+cx-{-dw-\-d-c. 

P= A ' 2> 

Example 1. If 3 gallons of wine at $1.37 per gallon, 2 at $2.18, and 5 at $1.75, 
be mixed together, what is a gallon worth of the mixture ? 
A = 3-f 2+5 = 10 gallons. 

_ 3Xl.37-f2X2.184-5Xl.T5 M wo 

P = — — — = $1.72 per gallon. 



Alligation.— Involution. 21 



Alligation of two ingredients a and b, with their respective prices or value per 
unit, z and y, z>P^>y. A = a+6. 

a : 6 = (P-y) : (z-P) 3, 

h{P-y ) . A (P-y) 

a = ~^Zpy anda = _— -, .... 4, & 5, 

Example 2. A Silver-smith will mix two sorts of silver, one at 54 and one at 
64 cents per ounce. How much must he taken of each sort to make the mixture 
worth 60 cents per ounce. (Formula 3.) P= 60. x = 54. y = 64. 
a : b = (60—54) : (64—60) = 6:4, or, 

4 ounces at 54 cents ; and 6 ounces at 64 cents. 

Alligation of three ingredients, a, b, and c, with their prices or value per unit, 
z. y and x. 

a' : J = (P—x): (z—P) 6, 

a" : b = (P—y) : (z— P) when *> P> y> a?, - - 7, 
6 : c" = (P—x) : (y—P) when *> y> P> *, - - 8, 
a = a'+a", c = d-\ c". 

Example 3. A Farmer will mix wheat at 94 cents per hushel, with barley 
at 72 cents, rye at 64 cents per hushel. How much of each sort must he taken 
to make the mixture worth 80 cents per hushel ? 

(Formula 6.) z = 94, y = 72, x = 64, and P = 80. 

a r : c = (80—64) : (94—80) = 16 : 14, 
a ' : b = (80—72) : (94—80) = 8 : 14. 
The wheat a = 16+8 = 24 bushels at 94 cents per hushel. 
" harley 6 = 14 " " 72 « « 

« rye c = 14 " " 64 " " 

Alligation of four ingredients a, b, c, and d, respective prices or value per unit ; 
z, y, x, and w. 

a' :d = (P— w) : (z—P) ) - ( 11, 

a" : b = (P—y) : {z—P) When *>P>y>r>w. - J 12, 
a'" : c = (P—x) : (*— P) J (^3, 

J 15, 
[16, 

In the same manner, formulae can he set up for any number of ingredients. 



a = a'-\-a"+a'" 
aid' =(P— w):(z— P)) 
b : d" = (P— w) : (y—P) When *>2/>aC>P>w. 
c : d>» = (P— to) : (x—P) J 
d = d'+d"+d w , 



INVOLUTION. 

Involution is to multiply a number into itself a number of times; each 
product is called the power of the number, and the dignity of the power is 
marked by a small figure called exponent, on the right of the number ; thus, 
aX« = a 2 the square of a. 
«X«Xa = a 3 the cube of a. 
«X«X«X« = a 4 the bisquare of a. 
aX^XaXaXa = a 5 the fifth power of a. 
<&., <fc., tfc. 

32 = 3X3 = 9. 
23 = 2X2X2 = 8. 
44 = 4X^X4X4 = 256. 
e&., dfc., efc. 

Binomc is a factor or quantity which contains two terms ; as (a-f-o.) 



22 Involution.— Evolution. 



Binomial-Theorem is the rule which a hinome follows, when it is raised 

to any power. 

When a binome is to be multiplied by itself or any other binome, it is set up 
and performed like the common multiplication by numbers ; thus, 

Exam * U1 - J#} (a+6)(a-f6), 

a 2 +ab 

aH2a6+b2 = (a+6) 2 . 

Example 2. Suppose a+b = 358746, and a = 358000, 6 = 746, then, 
(a+6)2 = 3587462, 

II II 

+a2 = 128164000000 
+2ao = 534136000 
+62 _ 556516 

128698692516 = (a+6)2. 

Ex. 3. (a+6)» = a8+3a25+3a62+6». 

. Ex. 4. {a+by = a*-f 4«3&-]-6a2&2 + 4 a &3+fc4. 
.Ec. 5. (a— 6)7 = a 7— 7a 6 6+21a562_35 a 453-f35a364— a265+7 a 6T— 67. 

Here you will discover the peculiarities of the BinomialrTheorem, which is thus 
expressed in words : 

1st. The exponent of the first term a in the power, is equal to the exponent 
of the binome ; and in every successive term, the exponent of a is decreased by 
1, until the last term of the exponent of a is 0, and, therefore, disappears, because 
any quantity raised to no power is equal to 1, thus, a = 1 and a 1 = a. 

2d. In the first term of the power, the exponent of 6 is 0, and therefore 6 will 
first appear in the second term with the exponent 1, and in every successive 
term the exponent of 6 is increased by I, until in the last term the exponent will 
be equal to the exponent of the binome. 

3d. The coefficient of the second term in the power, is equal to the exponent 
of the binome, and the coefficient of &ny successive term is equal to the product 
of the coeflicient and exponent of a in the foregoing term, divided by the num- 
ber of terms before the sought coefficient. 

4th. When the second term in the binome is negative, the first term in the 
power will be positive, the second negative, the third positive, the fourth nega- 
tive, &c, &c. The odd terms are positive, and the even terms are negative. 

5th. The number of terms in the power is one more than the exponent of the 
binome. 



EVOLUTION. 

Evolution is the reverse of Involution, or to find the number that has pro- 
duced a given power. In this case the given power is called the number, and 
the number which has produced the given power is called the root of the num- 
ber. The symbol y is generally placed over the number of which the root is 
to be extracted. The dignity of the root is placed thus $ of which the figure * 
is called the index of the root ; for the square roots the index 2 is always 
omitted. 
Example 1. \/~9j= 3 because 3 2 = 9. 

" 2. ff 6T=4 because 43 = 64. 

" 3. #531441=27 " 27* = 531441. 

cSc., <fc., 

In the accompanying Table are calculated the Squares, Cubes, Square Roots and 
Cube Hoots of any number up to 1600. By means of this Table, there will be easy 
rules to find the Square Root and Cube Root of numbers exceeding 1600. 



Table of Squares, Cubes, Square and Cube Roots. 



23 



Number. 


Squares. 


Cubes. 


\f Roots. 




Reciprocals. 


V Hoots. 


1 


1 


1 


1-0000000 


1-0000000 


•100000000 


2 


4 


8 


1-4142136 


1-2599210 


•500000000 


3 


9 


27 


1-7320508 


1-4422496 


•333333333 


4 


16 


64 


2-0000000 


1-5874011 


•250000000 


5 


25 


125 


2-2360680 


1-7099759 


•200000000 


6 


36 


216 


2-4494897 


1-8171206 


•166666667 


7 


49 


343 


2-6457513 


1-9129312 


•142S57143 


8 


64 


512 


2-8284271 


2-0000000 


•125000000 


9 


81 


729 


3-0000000 


2-0800837 


•111111111 


10 


100 


1000 


3-1622777 


2-1544347 


•100000000 


11 


121 


1331 


3-3166248 


2-2239801 


•090909091 


12 


144 


1728 


3-4641016 


2-2894286 


•083333333 


13 


169 


2197 


3-6055513 


2-3513347 


•076928077 


14 


196 


2744 


3-7416574 


2-4101422 


•071428571 


15 


225 


3375 


3-8729833 


2-4662121 


•066666667 


16 


256 


4096 


4-0000000 


2-5198421 


•062500000 


17 


289 


4913 


4-1231056 


2-5712816 


.058823529 


18 


324 


5832 


4-2426407 


2-6207414 


•055555556 


19 


361 


6859 


4-3588989 


2-6684016 


•052631579 


20 


400 


8000 


4-4721360 


2-7144177 


•050000000 


21 


441 


9261 


4-5825757 


2-7589243 


•047619048 


22 


484 


10648 


4-6904158 


2-8020393 


•045454545 


23 


529 


12167 


4-7958315 


2-8438670 


•043478261 


24 


576 


13824 


4-8989795 


2-8844991 


•041666667 


25 


625 


15625 


5-0000000 


2-9240177 


•040000000 


26 


676 


17576 


5-0990195 


2-9624960 


•038461538 


27 


729 


19683 


5-1961524 


3-0000000 


•037037037 


28 


784 


21952 


5-2915026 


3-0365889 


•035714286 


29 


841 


24389 


5-3851648 


3-0723168 


•034482759 


30 


900 


27000 


5-4772256 


3-1072325 


•033333333 


31 


961 


29791 


5-5677644 


3-1413806 


•032258065 


32 


1024 


32768 


5-6568542 


3-1748021 


•031250000 


33 


1089 


35937 


5-7445626 


3-2075343 


•030303030 


34 


1156 


39304 


5-8309519 


3-2396118 


•029411765 


35 


1225 


42875 


5-9160798 


3-2710663 


•028571429 


36 


1296 


46656 


6-0000000 


3-3019272 


•027777778 


»7 


1369 


50653 


6-0827625 


3-3322218 


•027027027 


38 


1444 


54872 


6-1644140 


3-3619754 


•026315789 


39 


1521 


59319 


6-2449980 


3-3912114 


•025641026 


40 


1600 


64000 


6-3245553 


3-4199519 


•025000000 


41 


1681 


68921 


6-4031242 


3-4482172 


•024390244 


42 


1764 


74088 


6-4807407 


3-4760266 


•023809524 


43 


1849 


79507 


6-5574385 


3-5033981 


•023255814 


44 


1936 


85184 


6-6332496 


3-5303483 


•022727273 


45 


2025 


91125 


6-7082039 


3-5568933 


•022222222 


46 


2116 


97336 


6-7823300 


3-5830479 


•021739130 


47 


2209 


103823 


6-8556546 


3-6088261 


•021276600 


48 


2304 


110592 


6-9282032 


3-6342411 


•020833333 


49 


2401 


117649 


7-0000000 


3-6593057 


•020408163 


50 


2500 


125000 


7-0710678 


3-6840314 


•020000000 


51 


2601 


132651 


7-1414284 


3-7084298 


•019607843 


52 


2704 


140608 1 


7-2111026 


3-7325111 


•019230769 



24 


Table of Squares, Cubes, Square 


and Cube Roots. 


Number. 


Squares. 


Cubes. 






Reciprocals. 


y/ Roots. 


\J Roots. 


53 


2809 


148877 


7-2801099 


3-7562858 


•018867925 


54 


2916 


157464 


7-3484692 


3-7797631 


•018518519 


55 


3025 


166375 


7-4161985 


3-8029525 


•018181818 


56 


3136 


175616 


7-4833148 


3-8258624 


•017857143 


57 


3249 


185193 


7*5498344 


3-8485011 


•017543860 


58 


3364 


195112 


7-6157731 


3-8708766 


•017241379 


59 


3481 


205379 


7-6811457 


3-8929965 


•016949153 


60 


3600 


216000 


7-7459667 


3-9148676 


•016666667 


61 


3721 


226981 


7-8102497 


3-9304972 


•016393443 


62 


3844 


238328 


7*8740079 


3-9578915 


•016129032 


63 


3969 


250047 


7-9372539 


3-9790571 


•015873016 


64 


4096 


262144 


8-0000000 


4-0000000 


•015625000 


65 


4225 


274625 


8-0622577 


4-0207256 


•015384615 


66 


4356 


287496 


8-1240384 


4-0412401 


•015151515 


67 


4489 


300763 


8-1853528 


4-0615480 


•014925373 


68 


4624 


314432 


8-2462113 


4-0816551 


•014705882 


69 


4761 


328509 


8-3066239 


4-1015661 


•014492754 


70 


4900 


343000 


8-3666003 


4-1212853 


•014285714 


71 


5041 


357911 


8-4261498 


4-1408178 


•014084517 


72 


5184 


373248 


8-4852814 


4-1601676 


•013888889 


73 


5329 


389017 


8-5440037 


4-1793390 


•013698630 


74 


5476 


405224 


8-6023253 


4-1983364 


•013513514 


75 


5625 


421875 


8-6602540 


4-2171633 


•013333333 


76 


5776 


438976 


8-7177979 


4-2358236 


•013157895 


77 


5929 


456533 


8-7749644 


4-2543210 


•012987013 


78 


6084 


474552 


8-8317609 


4-2726586 


•012820513 


79 


6241 


493039 


8-8881944 


4-2908404 


•012658228 


80 


6400 


512000 


8-9442719 


4-3088695 


•012500000 


81 


6561 


531441 


9-0000000 


4-3267487 


•012345679 


82 


6724 


551368 


9-0553851 


4-3444815 


•012195122 


83 


6889 


571787 


9-1104336 


4-3620707 


•012048193 


84 


7056 


592704 


9-1651514 


4-3795191 


•011904762 


85 


7225 


614125 


9-2195445 


4-3968296 


•011764706 


86 


7396 


636056 


9-2736185 


4-4140049 


•011627907 


87 


7569 


658503 


9-3273791 


4-4310476 


•011494253 


88 


7744 


681472 


9-3808315 


4-4470692 


•011363636 


89 


7921 


704969 


9-4339811 


4-4647451 


•011235955 


90 


8100 


729000 


9-4868330 


4-4814047 


•011111111 


91 


8281 


753571 


9-5393920 


4-4979414 


•010989011 


92 


8464 


778688 


9-5916630 


4-5143574 


•010869565 


93 


8649 


804357 


9-6436508 


4-5306549 


•010752688 


94 


8836 


830584 


9-6953597 


4-5468359 


•010638298 


95 


9025 


857374 


9-7467943 


4-5629026 


•010526316 


96 


9216 


884736 


9-7979590 


4-5788570 


•010416667 


97 


9409 


912673 


9-8488578 


4-5947009 


•010309278 


98 


9604 


941192 


9-8994949 


4-6104363 


•010204082 


99 


9801 


970299 


9-9498744 


4-6260650 


•010101010 


100 


10000 


1000000 


10-0000000 


4-64158S8 


•010000000 


101 


10201 


1030301 


10-0498756 


4-6570095 


•009900990 


102 


10404 


1061208 


10-0995049 


4-6723287 


•009803922 


103 


10609 


1092727 


10-1488916 


4-6875482 


•009708738 


104 


10816 


1124864 


10-1980390 


4-7025694 


•0096153S5 



Table of Squares, Cubes, Square and Cube Roots 



25 



Number. 


Squares. 


Cubes. 




V Roots. 


Reciprocals. 


\l Roots. 


105 


11025 


1157625 


10-2469508 


4-7176940 


•0095^810 


106 


11236 


1191016 


10-2956301 


4-7326235 


•009433962 


ior 


11449 


1225043 


10-3440804 


4-7474594 


•009345794 


108 


11664 


1259712 


10-3923048 


4-7622032 


•009259259 


109 


11881 


1295029 


10-4403065 


4-7768562 


•009174312 


110 


12100 


1331000 


10-4880885 


4-7914199 


•009090909 


111 


12321 


1367631 


10-5356538 


4-805S995 


•009009009 


112 


12544 


1404928 


10-5830052 


4-8202845 


•008928571 


113 


12769 


1442897 


10-6301458 


4-8345881 


•008849558 


114 


12996 


1481544 


10-6770783 


4-8488076 


•008771930 


115 


13225 


1520875 


10-7238053 


4-8629442 


•008695652 


116 


13456 


1560896 


10-7703296 


4-8769990 


•008020690 


117 


13689 


1601613 


10-8166538 


4-8909732 


•008547009 


118 


13924 


1643032 


10-8627805 


4-9048681 


•008474576 


119 


14161 


1685159 


10-9087121 


4-9186847 


•008403361 


120 


14400 


1728000 


10-9544512 


4*9324242 


•008333333 


121 


14641 


1771561 


11-0000000 


4-9460874 


•008264463 


122 


14834 


1815848 


11-0453610 


4-9596757 


•008196721 


123 


15129 


1860867 


11-0905365 


4-9731898 


•008130081 


124: 


15376 


1906624 


11-1355287 


4-9866310 


•008064516 


125 


15625 


1953125 


11-1803399 


5-0000000 


•008000000 


126 


15876 


2000376 


11-2249722 


5-0132979 


•007936508 


127 


16129 


2048383 


11-2694277 


5-0265257 


•007874016 


128 


16384 


2097152 


11-3137085 


5-0396842 


•007812500 


129 


16641 


2146689 


11-3578167 


5-0527743 


•007751938 


130 


16900 


2197000 


11-4017543* 


5-0657970 


•007692308 


131 


17161 


2248091 


11-4455231 


5-0787531 


•007633588 


132 


17424 


2299968 


11-4891253 


5-0916434 


•007575758 


133 


17689 


2352637 


11-5325626 


5-1044687 


•007518797 


134 


17956 


2406104 


11-575S369 


5-1172299 


•007462687 


135 


18225 


2460375 


11-6189500 


5-lJp9278 
5-1425632 


•007407407 


136 


18496 


2515-idQ 


11-6619038 


•007352941 


137 


18769 


2571353 


11-7046999 


5-1551367 


•007299270 


138 


19044 


2628072 


11-7473444 


5-1676493 


•007246377 


139 


19321 


2685619 


11-7S98261 


5-1801015 


•007194245 


140 


19600 


2744000 


11-8321596 


5-1924941 


•007142857 


141 


19S81 


2803221 


11-8743421 


5-2048279 


•007092199 


142 


20164 


2863288 


11-9163753 


5-2171034 


•007042254 


143 


20449 


2924207 


11-9582607 


5-2293215 


•006993007 


144 


20736 


2985984 


12-0000000 


5-2414828 


•006944444 


145 


21025 


3048625 


12-0415946 


5-2535879 


•006896552 


146 


21316 


3112136 


12-0830460 


5-2656374 


•006849315 


147 


21609 


3176523 


12-1243557 


5-2776321 


•006802721 


148 


21904 


3241792 


12-1655251 


5*2895725 


•006756757 


149 


22201 


3307949 


12-2065556 


5-3014592 


•006711409 


150 


22500 


3375000 


12-2474487 


5-3132928 


•006666667 


151 


22801 


3442951 


12-2882057 


5-3250740 


•006622517 


152 


23104 


3511008 


12-32S8280 


5-3368033 


•006578947 


153 


23409 


35S1577 


12-3693169 


5-3484812 


•006535948 


154 


23716 


3652264 


12-4096736 


5-3601084 


•006493506 


155 


24025 


3723575 


12-4498996 


5-3716854 


•006451613 


156 


24336 


3796416 


12-4899960 


5-3832126 


•006410256 



26 



Table of Squares, Cubes, Square and Cube Roots. 



Number. 


Squares. 


Cubes. 






Reciprocals. 


>/ Boots. 


f/ Roots. 


157 


24649 


3869893 


12-5299641 


5-3946907 


•006369427 


158 


24964 


3944312 


12-5698051 


5-4061202 


•006329114 


159 


25281 


4019679 


12-6095202 


5-4175015 


•006289308 


160 


25600 


4096000 


12-6491106 


5-4288352 


•006250000 


161 


25921 


4173281 


12-6885775 


5-4401218 


•006211180 


162 


26244 


4251528 


12-7279221 


5-4513618 


•006172840 


163 


26569 


4330747 


12-7671453 


5-4625556 


•006134969 


164 


26896 


4410944 


12-8062485 


5-4737037 


•006097561 


165 


27225 


4492125 


12-8452326 


5-4848066 


•006060606 


166 


27556 


4574296 


12-8840987 


5-4958647 


•006024096 


167 


27889 


4657463 


12-9228480 


5-5068784 


•005988024 


168 


28224 


4741632 


12-9614814 


5-5178484 


•005952381 


169 


28561 


4826809 


13-0000000 


5-5287748 


•005917160 


170 


28900 


4913000 


13-0384048 


5-5396583 


•005882353 


171 


29241 


5000211 


13-0766968 


5-5504991 


•005847953 


172 


29584 


5088448 


13-1148770 


' 5-5612978 


•005813953 


173 


29929 


5177717 


13-1529464 


5-5720546 


•005780347 


174 


30276 


5268024 


13-1909060 


5-5827702 


•005747126 


175 


30625 


5359375 


13-2287566 


5-5934447 


•005714286 


176 


30976 


5451776 


13-2664992 


5-6040787 


•005681818 


177 


31329 


5545233 


13-3041347 


5-6146724 


•005649718 


178 


31684 


5639752 


13-3416641 


5-6252263 


•005617978 


179 


32041 


5735339 


13-3790882 


5-6357408 


•005586592 


180 


32400 


5832000 


13-4164079 


5-6462162 


•005555556 


181 


32761 


5929741 


13-4536240 


5-6566528 


•005524862 


182 


33124 


6028568 


13-4907376 


5-6670511 


•005494505 


183 


33489 


6128487 


13-5277493 


5-6774114 


•005464481 


184 


33856 


6229504 


13-5646600 


5-6877340 


•005434783 


185 


34225 


6331625 


13-6014705 


5-6980192 


•005405405 


186 


34596 


6434856 


13-6381817 


5-7082675 


•005376344 


187 


34969 


6539203 


13-6747943 


5-7184791 


•005347594 


188 


35344 


66&672 


13-7113092 


5-7286543 


•005319149 


189 


35721 


6751269 


13-7477271 


5-7387936 


•005291005 


190 


36100 


6859000 


13-7840488 


5-7488971 


•005263158 


191 


36481 


6967871 


13-8202750 


5-7589652 


•005235602 


192 


36864 


7077888 


13-8564065 


5-7689982 


•005208333 


193 


37249 


7189517 


13-8924400 


5-7789966 


•005181347 


194 


37636 


7301384 


13-9283883 


5-7889604 


•005154639 


195 


38025 


7414875 


13-9642400 


5-7988900 


•005128205 


196 


38416 


7529536 


14-0000000 


5-8087857 


•005102041 


197 


38809 


7645373 


14-0356688 


5-8186479 


•005076142 


198 


39204 


7762392 


14-0712473 


5-8284867 


•005050505 


199 


39601 


7880599 


14-1067360. 


5-8382725 


•005025126 


200 


40000 


8000000 


14-1421356 


5-8480355 


•005000000 


201 


40401 


8120601 


14-1774469 


5-8577660 


•004975124 


202 


40804 


8242408 


14-2126704 


5-8674673 


•004950495 


203 


41209 


8365427 


14-2478068 


5-8771307 


•004926108 


204 


41616 


8489664 


14-2828569 


5-8867653 


•004901961 


205 


42025 


8615125 


14-3178211 


5-S9636S5 


•004878049 


206 


42436 


8741816 


14-3527001 


5-9059406 


•004854369 


207 


42849 


8869743 


14-3874946 


5-Q154817 


•004830918 


208 


43264 


8998912 


14-4222051 


5-9249921 


•004807692 



Table op Squares, Cubes, Square and Cube Roots. 



Number. 


Squares 


Cubes. 




V Koots. 


■ 


V Roots. 


Reciprocals. 


209 


436S1 


9129329 


14*4568323 


5-9344721 


-004784689 


210 


44100 


9261000 


14-4913767 


5-9439220 


-004761905 


211 


44521 


9393931 


14-525S390 


5-9533418 


•004739336 


212 


44944 


9528128 


14-5602198 


5-9627320 


•0047169SI 


213 


45369 


9663597 


14-5945195 


5-9720926 


•004694836 


214 


45796 


9800344 


14-6287388 


5-9814240 


•004672897 


215 


46225 


9938375 


14-6628783 


5-9907264 


•004651163 


216 


46656 


10077696 


14-6969385 


6-0000000 


•004629630 


217 


47089 


10218313 


14-7309199 


6-0092450 


•004608295 


218 


47524 


10360232 


14-7648231 


6-0184617 


•004587156 


219 


47961 


10503459 


14-7986486 


6-0276502 


•004566210 


220 


48400 


10648000 


14-8323970 


6-0368107 


•004545455 


221 


48841 


10793861 


14-8660687 


6-0459435 


•004524887 


222 


49284 


10941048 


14-8996644 


6-0550489 


•004504505 


223 


49729 


11089567 


14-9331845 


6-0641270 


•004484305 


224 


50176 


11239424 


14-9666295 


6-0731779 


•0044642S6 


225 


50625 


11390625 


15-0000000 


6-0824020 


•004444444 


226 


51076 


11543176 


15-0332964 


•6-0991994 


•004424779 


227 


51529 


11697083 


15-0665192 


6-1001702 


•004405286 


228 


51984 


11852352 


15-0996689 


6-1091147 


•004S85965 


229 


52441 


12008989 


15-1327460 


6-1180332 


•004366812 


230 


52900 


12167000 


15-1657509 


6-1269257 


•004347826 


231 


53361 


12326391 


15-1986842 


6-1357924 


•004329004 


232 


53824 


12487168 


15-2315462 


6-1446337 


•004310345 


233 


54289 


12649337 


15-2643375 


6-1534495 


•004291845 


234 


54756 


12812904 


15-2970585 


6-1622401 


•004273504 


235 


55225 


12977875 


15-3297097 


6-1710058 


•004255319 


236 


55696 


13144256 


15*3622915 


6-1797466 


•004237288 


237 


56169 


13312053 


15-3948043 


6-1884628 


•004219409 


238 


56644 


13481272 


15-4272486 


6-1971544 


•004201681 


239 


57121 


13651919 


15-4596248 


6-2058218 


•004184100 


240 


57600 


13824000 


15-4919334 


6-2144650 


•004166667 


241 


58081 


13997521 


15-5241747 


6-2230843 


•004149378 


242 


58564 


14172488 


15-5563492 


6-2316797 


•004132231 


243 


59049 


14348907 


15-5884573 


6-2402515 


•004115226 


244 


59536 


14526784 


15-6204994 


6-2487998 


•00409S361 


245 


60025 


14706125 


15-6524758 


6-2573248 


•0040S1633 


246 


60516 


14886936 


15-6S43871 


6-2658266 


•004065041 


247 


61009 


15069223 


15-7162336 


6-2743054 


•0040485S3 


248 


61504 


15252992 


15-7480157 


6-2827613 


•004032258 


249 


62001 


15438249 


15-7797338 


6-2911946 


•004016064 


250 


62500 


15625000 


15-8113883 


6-2996053 


•004000000 


251 


63001 


15813251 


15-8429795 


6-3079935 


•003984064 


252 


63504 


16003008 


' 15-8745079 


6-3163596 


•003968254 


253 


64009 


16194277 


15-9059737 


6-3247035 


•003952569 


254 


64516 


16387064 


15-9373775 


6-3330256 


•003937008 


255 


65025 


16581375 


15-9687194 


6-3413257 


•003921569 


256 


65536 


16777216 


16-0000000 


6-3496042 


•003906250 


257 


66049 


16974593 


16-0312195 


6-3578611 


•003891051 


258 


66564 


17173512 


16-0623784 


6-3660968 


•003875969 


259 


67081 


17373979 


16-0934769 


6-3743111 


•003861004 


260 


67600, 


17576000 


16-1245155 


6-3825043 


•003846154 



28 


iable op Squares, Cubes, Square 


a*st> Cube Roots. 


Number. 


Squares. 


Cubes. 




V Roots. 


Reciprocals. 


s/ Roots. 


261 


68121 


17779581 


16-1554944 


6-3906765 


•003831418 


262 


68644 


17984728 


16*1864141 


6-3988279 


•003816794 


263 


69169 


18191447 


16-2172747 


6-4069585 


•0038022S1 


264 


69696 


18399744 


16-2480768 


6-4150687 


•003787879 


265 


70225 


1S609625 


16-278S206 


6-4231583 


•003773585 


266 


70756 


1SS21096 


16-3095064 


6-4312276 


•003759398 


267 


71289 


19034163 


16-3401346 


6-4392767 


•003745318 


268 


71824 


1924S832 


16-3707055 


6-4473057 


•003731343 


269 


72361 


19465109 


16-4012195 


6-4553148 


•003717472 


270 


72900 


19683000 


16.4316767 


6-4633041 


•003703704 


271 


73441 


19902511 


16-4620776 


6-4712736 . 


•003690037 


272 


73984 


20123643 


16-4924225 


6-4792236 


•003676471 


273 


74529 


20346417 


16-5227116 


6-4871541 


•003663004 


274 


75076 


20570824 


16-5529454 


6-4950653 


•003649635 


275 


75625 


20796875 


16-5831240 


6-5029572 


•003636364 


276 


76176 


21024576 


16-6132477 


6-5108300 


•003623188 


277 


76729 


21253933 


16-6433170 


6-5186839 


•003610108 


278 


77284 


214849^2 


16-6783320 


6-5265189 


•003597122 


279 


77841 


21717639 


16-7032931 


6-5343351 


•003584229 


2S0 


78400 


21952000 


16-7332005 


6-5421326 


•003571429 


2S1 


78961 


22188041 


16-7630546 


6-5499116 


•00355S719 


282 


79524 


22425768 


16-7928556 


6-5576722 


•003546099 


283 


80089 


226651S7 


16-8226038 


6-5654144 


•003533569 


284 


80656 


22906304 


16-8522995 


6-5731385 


•003522127 


285 


81225 


23149125 


16-SS19430 


6-5808443 


•003508772 


286 


81796 


23393656 


16-9115345 


6-58S5323 


•003496503 


287 


82369 


23639903 


16-9410743 


6-5962023 


•003484321 


288 


82944 


23887872 


16-9705627 


6-6038545 


•003472222 


289 


83521 


24137569 


17-0000000 


6-6114890 


•003460208 


290 


84100 


24389000 


17-0293864 


6-6191060 


•00344S276 


291 


84681 


24642171 


17-0587221 


6-6267054 


•003436426 


292 


85264 


24897088 


17-08S0075 


6-6342874 


•003424658 


293 


85849 


25153757 


17-1172428 


0-6418522 


•003412969 


294 


86436 


254121S4 


17-1464282 


6-6493998 


•003401361 


295 


87025 


25672375 


17-1755640 


6-6569302 


•003389831 


296 


87616 


25934836 


17-2046505 


6-6644437 


•003378378 


297 


88209 


26198073 


17-2336879 


6-6719403 


•003367003 


298 


88804 


26463592 


17-2626765 


6.6794200 


•003355705 


299 


89401 


26730899 


17-2916165 


6.6S68831 


•003344482 


300 


90000 


27000000 


17-3205081 


6.6943295 


•003333333 


301 


90601 


27270901 


17-3493516 


6.7017593 


•003322259 


302 


91204 


27543608 


17-3781472 


6-7091729 


•003311258 


303 


91809 


27818127 


17-4068952 


6-7165700 


•003301330 


304 


92416 


28094464 


17-4355958* 


6-7239508 


•003289474 


305 


93025 


28372625 


17-4642492 


6-7313155 


•003278689 


306 


93636 


28052616 


17-4928557 


6-7386641 


•003207974 


307 


94249 


28934443 


17-5214155 


6-7459967 


•003257329 


308 


94S64 


29218112 


17-5499288 


6-7533134 


•003246753 


309 


95481 


29503609 


17*5783958 


6-7606143 


•003236246 


310 


96100 


29791000 


17-6068169 


6-7678995 


•003225806 


311 


96721 


300S0231 


17-6351921 


6-7751690 


•003215434 


312 


97344 


30371328 


17-6635217 


6-7824229 


•003205128 



Table of Squares, Cubes, Squaee and Cube Roots. 



23 



Number. 


Squares. 


Cubes. 




V Roots. 


Reciprocals. 


y/ Roots. 


313 


97969 


30664297 


17-6918060 


6-7S96613 


•003194883 


314 


98596 


30959144 


17-7200451 


6-7968844 


•003184713 


315 


99225 


31255875 


17-7482393 


6-8040921 


•003174603 


315 


99S56 


31554496 


17-7763883 


6-8112847 


•003164557 


317 


100489 


31855013 


17-8044938 


6-8184620 


.003154574 


318 


101124 


32157432 


17-8325545 


6-8256242 


•003144654 


319 


101761 


32461759 


17-8605711 


6-8327714 


•003134796 


320 


102400 


32768000 


17-8885438 


6-8399037 


•003125000 


321 


J03041 


33076161 


17-9164729 


6-8470213 


•003115265 


322 


103684 


33386248 


17-9443584 


6-8541240 


•003105590 


323 


104329 


33698267 


17-9722008 


6-8612120 


•003095975 


324 


104976 


34012224 


18-0000000 


6-8682855 


•003086420 


325 


105625 


34328125 


18-0277564 


6-8753433 


•003076923 


326 


106276 


34645976 


18-0554701 


6-8823888 


•003067485 


327 


106929 


34965783 


1S-0831413 


6-8894188 


•003048104 


328 


107584 


35287552 


18-1107703 


6-S964345 


•003048780 


329 


108241 


35611289 


18-1383571 


6-9034359 


•003039514 


330 


10S900 


35937000 


18-1659021 


6-9104232 


•003030303 


331 


109561 


36264691 


18-1934054 


6*9173964 


•003021148 


332 


110224 


36594368 


18-2208672 


6-9243556 


•003.012048 


333 


110889 


36926037 


18-2482876 


6-9313088 


•003003003 


334 


111556 


37259704 


18-2756659 


6-9382321 


•002994012 


335 


112225 


37595375 


18-3030052 


6-9451496 


•002985075 


333 


112896 


37933056 


18-3303028 


6-9520533 


•002976190 


337 


113569 


38272753 


18-3575598 


6-95S9434 


•002967359 


338 


114244 


3S614472 


18-3847763 


6-9658198 


•002958580 


339 


114921 


3S958219 


18-4119526 


6-9726826 


•002949353 


340 


115600 


39304000 


18-4390889 


6-9795321 


;002941176 


341 


116281 


39651821 


18-4661853 


6-9863681 


•002932551 


342 


116964 


40001688 


18-4932420 


6-9931906 


•002923977 


343 


117649 


40353607 


18-5202592 


7-0000000 


•002915452 


344 


118336 


40707584 


18-5472370 


7-0067962 


•002906977 


345 


119025 


41063625 


18-5741756 


7-0135791 


•002S98551 


346 


119716 


41421736 


18-6010752 


7-0203490 


•002890173 


347 


120409 


41781923 


18-6279360 


7-0271058 


•002881844 


348 


121104 


42144192 


18-6547581 


7-0338497 


•002873563 


349 


121801 ! 4250S549 


18-6815417 


7-0405860 


•002865330 


350 


122500 42875000 


18-7082869 


7-0472987 


•002857143 


351 


123201 43243551 


18-7349940 


7-0540041 


•002849003 


352 


123904 


43614208 


18-7616630 


7-0606967 


•002840909 


353 


124609 


43986977 


18-7882942 


7-0673767 


•002832861 


354 


125316 


44361864 


18-8148877 


7-0740440 


•002824859 


355 


126025 


44738875 


18-8414437 


7*0806983 


•002816901 


356 


126736 


45118016 


*18-8679623 


7-0873411 


•002808989 


357 


127449 


45499293 


18-8944436 


7-0939709 


•002801120 


358 


128164 


45882712 


18-920S879 


7-1005885 


•002793296 


359 


128881 


46268279 


18-9472953 


7-1071937 


•002785515 


360 


129600 


46656000 


18-9736660 


7-1137866 


•002777778 


361 


130321 


47045831 


19-0000000 


7-1203674 


•002770083 


362 


131044 


47437928 


19-0262976 


7-1269360 


•002762431 


363 


131769 


47832147 


19-0525589 


7-1334925 


•002754821 


364 


132496 


48228544 


19-0787840 


7-1400370 , 


•002747253 



3* 



BO 



Table op Squares, Cubes, Square and Cube Roots. 



Number. 


Squares' 


Cubes. 






Reciprocals. 


V^ Roots. 


V Roots. 


365 


133225 


! 48627125 


19-1049732 


7-1465695 


•002739726 


366 


133956 


1 49027896 


19-1311265 


7-1530901 


•002732240 


367 


134689 


: 49430863 


19-1572441 


7-1595988 


•002724796 


368 


135424 


: 49836032 


19-1833261 


7-1660957 


•002717393 


369 


136161 


j 50243409 


19-2093727 


7-1725809 


•002710027 


370 


136900 


150653000 


19-2353841 


7-1790544 


•002702703 


371 


137641 


51064811 


19-2613603 


7-1855162 


•002095418 


372 


138384 


f51478S48 


19-2873015 


7-1919663 


•002688172 


373 


139129 


! 51895117 


19-3132079 


7-1984050 


•002680965 
•002673797 


374 


139876 


52313624 


19-3390796 


7-2048322 


375 


140625 


52734375 


19-3619167 


7-2112479 


•002606667 


376 


141376 


53157376 


19-3907194 


7-2176522 


•002659574 


377 


142129 


535S2633 


19-4164878 


7-2240450 


•002652520 


378 


142SS4 


54010152 


19-4422221 


7-2304268 


•002645503 


379 


143641 


54439939 


19-4679223 


7-2367972 


•002638521 


380 


144400 


54872000 


19-4935887 


7-2431565 


•002631579 


381 


145161 


55306341 


19-5192213 


7-2495045 


•002624672 


382 


145924 


55742968 


19-5448203 


7-255S415 


•002617801 


383 


146689 


56181887 


19-5703858 


7-2621675 


•002610966 


384 


147456 


56623104 


19-5959179 


7-2684824 


•002604167 


385 


148225 


57066625 


19-6214169 


7-2747864 


•002597403 


386 


148996 


57512456 


19-6468827 


7-2810794 


•002590674 


387 


149769 


57960603 


19-6723156 


7-2873617 


•002583979 


388 


150514 


58411072 


19-6977156 


7-2936330 


•002577320 


389 


151321 


58S63869 


19-7230829 


7-2998936 


•002570694 


390 


152100 


59319000 


19-7484177 


7-3061436 


•002564103 


391 


152S81 


59776471 


19-7737199 


7-3123828 


•002557545 


392 


1^3664 


60236288 


19-79S9S99 


7-31S6114 


•002551020 


393 


154449 


60698457 


19-8242276 


7-324S295 


•002544529 


394 


155236 


61162984 


19-8494332 


7-3310369 


•002538071 


395 


156025 


61629875 


19-8746069 


7-3372339 


•002531646 


396 


156816 


62099136 


19-8997487 


7-3434205 


•002525253 


397 


157609 


62570773 


19-9248588 


7-3495966 


•002518892 


398 


15S404 


63044792 


19-9499373 


7-3557624 


•002512563 


399 


159201 


63521199 


19-9749844 


7-3619178 


•002506266 


400 


160000 


64000000 


20-0000000 


7-3680630 


•002500000 


401 


160801 


64481201 


20-0249844 


7-3741979 


•002493766 


402 


161604 


64964S08 


20-0499377 


7-3803227 


•002487562 


403 


162409 


65450827 


20-0748599 


7-3864373 


•002481390 


404 


163216 


65939264 


20-0997512 


7-3925418 


•00247524S 


405 


164025 


66430125 


20-1246118 


7-3986363 


• -002469136 


406 


164836 


06923416 


20-1494417 


7-4047206 


•002463054 


407 


165649 


67419143 


20-1742410 


7-4107950 


•002457002 


408 


166464 


67917312 


20-1990099 


7-4168595 


•002450980 


409 


167281 


68417929 


20-2237484 


7-4229142 


•002444988 


410 


168100 


68921000 


20-2484^67 


7-4289589 


•002439024 


411 


168921 


69426531 


20-2731349 


7-4349938 


•002433090 


412 


169744 


69934528 


20-2977831 


7-4410189 


•002427184 


413 


170569 


70444997 


20-3224014 


7-4470343 


•002421308 


414 


171396 


70957944 


20-3469899 


7-4530399 


•002415459 


415 


172225 


71473375 


20-3715488 


7-4590359 


•002409639 


41S 1 


17305C 


719912961 


20-3960781 


7-4650223 


•002406846 





Table of Squares 


, Cubes, Square 


and Cube Roots. 31 


Number. 


I 

j Squares. 


Cubes. 


V Roots. 




Reciprocals. 


\/ Roots. 


417 


1738S9 


72511713 


20-4205779 


7-4709991 


•002398082 


418 


174724 


73034632 


20-4450483 


7-4769664 


•002392344 


419 


175561 


73560059 


20-4694895 


7-4829242 


•002386635 


420 


176400 


74088000 


20-4939015 


7-4SS8724 


•002380952 


421 


177241 


74618461 


20-5182845 


7-4948113 


•002375297 


422 


1780S4 


75151448 


20-5426386 


7-5007406 


•002369668 


423 


178929 


75686967 


20-5669638 


7-5066607 


•002364066 


424 


179776 


76225024 


20-5912603 


7-5125715 


•002358491 


425 


180625 


76765625 


20-6155231 


7-5184730 


•002352941 


42 G 


181476 


77308776 


20-6397674 


7-5243652 


•002347418 


• 427 


182329 


77854483 


20-6639783 


7*5302482 


•002341920 


428 


183184 


78402752 


20-8881609 


7-5361221 


•002336449 


42% 


184041 


78953589 


20-7123152 


7-5419867 


•002331002 


430 


184900 


79507000 


20-7364414 


7*5478423 


•002325581 


431 


185761 


80062991 


20-7605395 


7*5536888 


•002320186 


432 


186624 


80621568 


20-7846097 


7*5595263 


•002314815 


433 


187489 


81182737 


20-8086520 


7-5653548 


•002309469 


434 


188356 


81746504 


20-8326667 


7-5711743 


•002304147 


435 


189225 


82312875 


20-8506536 


7*5769849 


•002298851 


436 


190096 


82S81856 


20-8806130 


7-5827865 


•002293578 


437 


190969 


83453453 


20-9045450 


7-5885793 


•002288330 


438 


191844 


84027672 


20-9284495 


7-5943633 


•002283105 


439 


192721 


84604519 


20-9523268 


7-6001385 


•002277904 


440 


193600 


85184000 


20-9761770 


7-6059049 


•002272727 


,441 


194481 


85766121 


21-0000000 


7-6116626 


•002267574 


442 


195364 


86350888 


21-0237960 


7-6174116 


•002262443 


443 


196249 


86938307 


21-0475652 


7-6231519 


•002257336 


444 


197136 


87528384 


21-0713075 


7-6288837 


•002252252 


445 


198025 


88121125 


21-0950231 


7-6346067 


•002247191 


446 


198916 


88716536 


21-1187121 


7-6403213 


•002242152 


.447 


199809 


89314623 


21-1423745 


7-6460272 


•002237136 


448 


200704 


89915392 


21-1660105 


7-6517247 


•002232143 


449 


201601 


90518849 


21-1896201 


7-6574138 


•002227171 


450 


202500 


91125000 


21-2132034 


7-6630943 


•002222222 


451 


203401 


91733851 


21-2367606 


7*6687665 


•002217295 


452 


204304 


92345408 


21-2602916 


7-6744303 


•0022123S9 


453 


205209 


92959677 


21-2837967 


7-6800857 


•002207506 


454 


206116 


93576664 


21-3072758 


7-6857328 


•002202643 


455 


207025 


94196375 


21-3307290 


7-6913717 


•002197802 


456 


207936 


94818816 


21-3541565 


7-6970023 


•002192982 


457 


208849 


95443993 


21-3775583 


7-7026246 


•002188184 


458 


209764 


96071912 


21-4009346 


7-7082388 


•002183406 


459 


210681 


96702579 


21-4242853 


r-7188448 


•002178649 


460 


211600 


97336000 


21-4476106 


7*7194426 


•002173913 


461 


212521 


97972181 


21-4709106 


7*7250325 


•002169197 


462 


213444 


98611128 


21-4941853 


7*7306141 


•002164502 


463 


214369 


99252847 


21-5174348 


7-7361877 


•002159827 


464 


215296 


99897344 


21-5406592 


7-7417532 


•002155172 


465 


216225 


100544625 


21-5838587 


7-7473109 


•002150538 


466 


217156 


101194696 


21-5870331 


7-7528606 


•002145923 


467 


218089 


101847563 


21-6101828 


7*7584023 


•002141328 


468 


219024 


102503232 


21-6333077 1 


7*7639361 


•002136752 



32 



Table op Squares. Cubes, Square and Cube Roots. 



Number. 


-». . . , 








I Reciprocals. 


Squares. 


Cubes. 


\/ Roots. 


4/ Roots. 


469 


219961 


103161709 


21-6564078 


7-7694620 


•002132196 


470 


220900 


103S23000 


21-6794834 


7-7749801 


•002127660 


471 


221841 


104487111 


21-7025344 


7-7804904 


•002123142 


472 


222784 


105154048 


21-7255610 


7-7859928 


•00211S644 


473 


223729 


105828817 


21-7485632 


7*7914875 


•002114165 


474 


224676 


106496424 


21-7715411 


7*7969745 


•002109705 


475 


225625 


107171875 


21-7944947 


7-8024538 


•002105263 


476 


226576 


107850176 


21-8174242 


7-8079254 


•002100840 


477 


227529 


108531333 


21*8403297 


7-8133892 


•002096486 


478 


228484 


109215352 


21-8632111 


7-8188456 


•002092050 


479 


229441 


109902239 


21-8860686 


7*8242942 


•002087683 


480 


230400 


110592000 


21-9089023 


7-8297353 


•002083333 


481 


231361 


111284641 


21-9317122 


7-8351688 


•002079002 


482 


232324 


1119S0168 


21-9544984 


7-8405949 


•0020?4689 


483 


233289 


112678587 


21-9772610 


7-8460134 


•002070393 


484 


234256 


113379904 


22-0000000 


7-8514244 


•002066116 


485 


235225 


114084125 


22-0227155 


7-8568281 


•002061856 


486 


236196 


114791256 


22-0454077 


7-8622242 


•002057613 


487 


237169 


115501303 


22-0680765 


7-8676130 


•002053388 


488 


238144 


116214272 


22-0907220 


7-S729944 


•002049180 


489 


239121 


116930169 


22-1133444 


7-8783684 


•002044990 


490 


240100 


117649000 


22-1359436 


7-8837352 


•002040816 


491 


241081 


118370771 


22-1585198 


7-S890946 


•002036660 


492 


242064 


119095488 


22-1810730 


7-8944468 


•002032520 


493 


243049 


119823157 


22-2036033 


7-8997917 


•00202S398 


494 


244036 


120553784 


22-2261108 


7-9051294 


•002024291 


495 


245025 


121287375 


22-24S5955 


7-9104599 


•002020202 


496 


246016 


122023936 


22-2710575 


7-9157832 


•002016129 


497 


247009 


122763473 


22-2934968 


7-9210994 


•002012072 


498 


248004 


123505992 


22-3159136 


7-9264085 


•002008032 


499 


249001 


124251499 


22-3383079 


7-9317104 


•002004008 


500 


250000 


125000000 


22-3606798 


7-9370053 


•002000000 


501 


251001 


125751501 


22-3830293 


7-9422931 


•001996008 


502 


252004 


126506008 


22-4053565 


7-9475739 


•001992032 


503 


253009 127263527 


22-4276615 


7-9528477 


•001988072 


504 


254016 128024064 


22-4499443 


7*9581144 ' 


•001984127 


505 


255025! 128787625 


22-4722051 


7-9633743 


•001980198 


506 


256036! 129554216 


22-4944438 


7-9686271 


•001976285 


507 


257049 


130323843 


22-5166605 


7-9738731 


•001972387 


508 


258064 


131096512 


22-5388553 


7-9791122 


•001968504 


509 


259081 


131872229 


22-5610283 


7-9843444 


•001964637 


510 


260100 


132651000 


22-5831796 


7-9895697 


•001960784 


511 


261121 


133432831 


22-6053091 


7-9947883 


•001956947 


512 " 


262144 


134217728 


22-6274170 


8-0000000 


•001953125 


513 


263169 


135005697 


22-6495033 


8-0052049 


•001949318 


514 


264196 


135796744 


22-6715681 


8-0104032 


•001945525 


515 


265225 


136590875 


22-6936114 


8-0155946 


•001941748 


516 


266256 137388096 


22-7156334 


8-0207794 


•001937984 


517 


267289 


138188413 


22-7376341 


8-0259574 


•001934236 


518 


268324 


138991832 


22-7596134 


8-0311287 


•001930502 


519 


269361 


139798359 


22-7815715 


8-0362935 


•001926782 


520 1 


270400 140608000 


22-8035085 


8-0414515 1 


•001923077 



■ * 


Table of Squares, 


Ctjbes, Square 


and Cube Roots. 


S3 


Number, 


Squares. 


Cubes. 


V^Roots. 


4/ Roots. 1 


leciprocals. 


521 


271411 


141420761 


22-8254244 


8-0466030 


•001919386 


522 


272484 | 14223GC48 


22-8473193 


S-0517479 


•001915709 


523 


273529 143055667 


22-8691933 


8-0568862 


•001912046 


524 


274576' 143877824 


22-8910463 


8-0620180 


•001908397 


525 


275625 1 144703125 


22-9128785 


8-0671432 


001904762 


526 


276676 | 145531576 


22-9346899 


8-0722620 


001901141 


527 


277729 146363183 


22-9564806 


8*0773743 * 


001897533 


528 


278784 


147197952 


22-9782506 


8-0824800 


001893939 


529 


279841 


148035889 


23-0000000 


8-0875794 


001890359 


530 


280900 


148877001 


23-0217289 


8-0926723 


001886792 


531 


281961 


149721291 


23-0434372 


8-0977589 


001883239 


532 


283024 


150568768 


23-0651252 


8-102S390 


001879699 


533 


284089 


151419437 


23-0867928 


8-1079128 


001876173 


534 


285156 


152273304 


23-1084400 


8-1129803 


001872659 


535 


286225 


153130375 


23-1300670 


8-1180414 


001869159 


536 


287296 


153990656 


23-1516738 


8-1230962 


001865672 


537 


288369 


154854153 


23-1732605 


8-1281447 


001862197 


538 


289444 


155720872 


23-1948270 


8-1331870 


001858736 


539 


290521 


156590819 


23-2163735 


8-1382230 


001855288 


540 


291600 


157464000 


23-2379001 


8-1432529 


001851852 


541 


292681 


i 58340421 


23-2594067 


8-1482765 


001848429 


542 


293764 


159220088 


23-2808935 


8-1532939 


001845018 


543 


294849 


160103007 


23-3023604 


8-1583051 


001841621 


544 


295936 


160989184 


23-323S076 


8-1633102 


001838235 


545 


297025 


161878625 


23-3452351 


8-1683092 


001834862 


546 


298116 


162771336 


23-3666429 


8-1733020 


001831502 


547 


299209 


163667323 


23-3880311 


8-1782888 


001828154 


548 


300304 


164566592 


23-4093998 


8-1832695 


001824818 


549 


301401 


165469149 


23-4307490 


8-1882441 


001821494 


550 


302500 


166375000 


23-4520788 


8-1932127 


001818182 


551 


303601 


167284151 


23-4733892 


8-1981753 


001814882 


552 


304704 


168196608 


23-4946S02 


8-2031319 


001811594 


553 


305809 


169112377 


23-5159520 


8-2080825 


001808318 


554 


306916 


170031464 


23-5372046 


8-2130271 


001805054 


555 


308025 


170953875 


23-5584380 


8-2179657 


001801802 


556 


309136 


171879616 


23-5796522 


8-2228985 


001798561 


557 


310249 


172S0S693 


23-6008474 


8-2278254 


001795332 


558 


311364 


173741112 


23-6220236 


8-2327463 


001792115 


559 


312481 


174676S79 


23-6431808 


8-2376614 


001788909 


560 


313600 


175616000 


23-6643191 


8-2425706 


001785714 


561 


314721 


176558481 


23-6854386 


8-2474740 


001782531 


562 


315844 


177504328 


23-7065392 


8-2523715 


001779359 


563 


316969 


178453547 


23-7276210 


8-2572635 


001776199 


564 


318096 


179406144 


23-7486842 


8-2621492 


001773050 


565 


319225 


180362125 


23-7697286 


8-2670294 


001769912 


566 


320356 


181321496 


23-7907545 


8-2719039 


001766784 


567 


321489 


182284263 


23-S117618 


8-2767726 


001763668 


508 


322624 


183250432 


23-8327506 


8-2816255 


001760563 


569 


323761 


184220009 


23-8537209 


8-2864928 


001757469 


570 


324900 


185193000 


23-8746728 


8-2913444 


001754386 


571 


326041 


186169411 


23-8956063 


8-2961903 


001751313 


572 


327184 187149248 23-9165215 1 


8-3010304 


001748252 








C 







34 


Table of Squares 


, Cubes, Square 


axd Cube Pi got 


s. 


Number. 


Squares. 


Cubes. 






Reciprocals. 


*J Roots. 


V Roots. 


573 


328329 


188132517 


23-9374184 


8-3058651 


•001745201 


574 


329476 


189119224 


23-9582971 


8-3106941 


•001742160 


575 


330625 


190109375 


23-9791576 


8-3155175 


•001739130 


576 


331776 


191102976 


24-0000000 


8-3203353 


•001736111 


577 


332927 


192100033 


24-0208243 


8-3251475 


•001733102 


578 


334084 


193100552 


24-0416306 


8-3299542 


•001730104 


•579 


335241 


194104539 


24-0624188 


8-3347553 


•001727116 


580 


336400 


195112000 


24-0831891 


8-3395509 


•001724138 


581 


337561 


196122941 


24-1039416 


8-3443410 


•001721170 


582 


338724 


197137368 


24-1246762 


8-3491256 


•001718213 


583 


339889 


198155287 


24-1453929 


8-3539047 


•C01715266 


584 


341056 


199176704 


24-1660919 


8-3586784 


•001712329 


585 


342225 


200201625 


24-1867732 


8-3634466 


•001709402 


586 


343396 


201230056 


24-2074369 


8-3682095 


•0017064S5 


587 


344569 


202262003 


24-2280829 


8-3729668 


•001703578 


588 


345744 


203297472 


24-2487113 


8-3777188 


•001700680 


589 


346921 


204336469 


24-2693222 


8-3824653 


•001697793 


590 


348100 


205379000 


24-2899156 


8-3872065 


•001694915 


591 


349281 


206425071 


24-3104996 


8-3919428 


•001692047 


592 


350464 


207474688 


24-3310501 


8-3966729 


•001689189 


593 


351649 


208527857 


24-3515913 


8-4013981 


•001686341 


594 


352836 


209584584 


24-3721152 


8-4061180 


•001683502 


595 


354025 


210644875 


24-3926218 


8-4108326 


•001680672 


596 


355216 


211708736 


24-4131112 


8-4155419 


•001677852 


597 


356409 


212776173 


24-4335834 


8-4202460 


•001675042 


598 


357604 


213847192 


24-4540385 


8-4249448 


•001672241 


599 


358801 


214921799 


24-4744765 


8-4296383 


•001669449 


600 


360000 


216000000 


24-4948974 


8-4343267 


•001666667 


601 


361201 


217081801 


24-5153013 


8-4390098 


•001663894 


602 


362404 


218167208 


24-5356883 


8-4430877 


•001661130 


603 


363609 


219256227 


24-5560583 


8-4483605 ' 


•001658375 


604 


364816 


220348864 


24-5764115 


8-4530281 


•001655629 


605 


366025 


221445125 


24-5967478 


8-4576906 


•001652893 


606 


367236 


222545016 


24-6170673 


8-4623479 


•001650165 


607 


368449 


223648543 


24-6373700 


8-4670001 


•001647446 


608 


369664 


224755712 


24-6576560 


8-4716471 


•001644737 


609 


370881 


225866529 


24-6779254 


8-4762892 


•001642036 


610 


372100 


226981000 


24-6981781 


8-4809261 


•001639344 


611 


373321 


228099131 


24-7184142 


8-4855579 


•001636661 


612 


374544 


229220928 


24-73S6338 


8-4901848 


•001633987 


613 


375769 


230346397 


24-7588368 


8-4948005 


•001631321 


614 


376996 


231475544 


24-7790234 


8-4994233 


•001628664 


615 


378225 


23260S375 


24-7991935 


8-5040350 


•001626016 


616 


379456 


233744896 


24-S;93473 
24-8394847 


8-5086417 


•001623377 


617 


380689 


234885113 


8-5132435 


•001620746 


618 


381924 


236029032 


24-8596058 


8-5178403 


•001618123 


619 


383161 


237176659 


24-8797106 


8-5224331 


•001615509 


620 


384400 


238328000 


24-8997992 


8-5270189 


•001612903 


621 


385641 


239483061 


24-9198716 


8-5316009 


•001610306 


622 


386884 


240641848 


24-9399278 


8-5361780 


•001607717 


623 


388129 


241804367 


24-9599679 


8-5407501 


•001605136 


624 


389376) 


242970624 


24-9799920 


8-5453173 


001602564 





Table op Squares 


Cubes, Square and Cube Roots. 35 


Number. 






4/ Roots. 


Reciprocals. 


Squares. 


Cubes. 


v/Boots. 


625 


390625 


244140625 


25-0000000 


8-5498797 


•001600000 


626 


391876 


245134376 


25-0199920 


8-5544372 


•001597444 


627 


393129 


246491883 


25-0399681 


8-5589899 


•001594S96 


628 


394384 


247673152 


25*0599282 


8-5635377 


•001592357 


629 


395641 


248858189 


25-0798724 


8-5680807 


•001589825 


630 


396900 


250047000 


25-0998008 


8-5726189 


•001587302 


631 


398161 


251239591 


25-1197134 


8-5771523 


•0015S47S6 


632 


399424 


252435968 


25-1396102 


8-5816809 


•001582278 


633 


4006S9 


253636137 


25*1594913 


8-5862247 


•001579779 


634 


401956 


254840104 


25-1793566 


8-5907238 


•001577287 


635 


403225 


256047875 


25-1992063 


8-5952380 


•001574803 


636 


404496 


25725*9456 


25-2190404 


8-5997476 


•001572327 


637 


405769 


258474853 


25-2388589 


8-6042525 


•001569S59 


638 


407044 


259694072 


25-2586619 


8-6087526 


•001567398 


639 


408321 


260917119 


25-2784493 


8-6132480 


•001564945 


640 


409600 


262144000 


25-2982213 


8-6177388 


•001562500 


641 


410881 


263374721 


25-3179778 


8-6222248 


•001560062 


642 


412164 


264609288 


25-33771S9 


8-6267063 


•001557632 


643 


413449 


265847707 


25-3574447 


8-6311830 


•001555210 


644 


414736 


267089984 


25-3771551 


8-6356551 


•001552795 


645 


416125 


268336125 


25-3968502 


8-6401226 


•001550388 


646 


417316 


269585136 


25-4165302 


8-6445855 


•001547988 


647 


418609 


270840023 


25-4361947 


8-6490437 


•001545595 


648 


419904 


272097792 


25-4558441 


8-6534974 


•001543210 


649 


421201 


273359449 


25*4754784 


8-6579465 


•001540S32 


650 


422500 


274625000 


25-4950976 


8-6623911 


•001538462 


651 


423801 


275894451 


25-5147013 


8-6668310 


•001536098 


652 


425104 


277167808 


25-5342907 


8-6712665 


•001533742 


653 


426409 


278445077 


25-5538647 


8-6756974 


•001531394 


654 


427716 


279726264 


25-5734237 


8-6801237 


•001529052 


655 


429025 


281011375 


25-5929678 


8-6845456 


•001526718 


656 


430336 


282300416 


25-6124969 


8-6889630 


•001524390 


657 


431639 


283593393 


25-6320112 


8-6933759 


•001522070 


658 


432964 


284890312 


25-6515107 


8-6977843 


•001519751 


659 


434281 


286191179 


25-6709953 


8-7021882 


•001517451 


660 


435600 


287496000 


25-6904652 


8-7005877 


•001515152 


661 


436921 


288804781 


25.7099203 


8-7109827 


•001512859 


662 


438244 


290117528 


25-7293607 


8-7153734 


•001510574 


663 


439569 


291434247 


25-7487864 


8-7197596 


•00150S296 


664 


440896 


292754944 


25-7681975 


8-7241414 


•001506024 


665 


442225 


294079625 


25-7875939 


8-72S5187 


•001503759 


666 


443556 


295408296 


25-8069758 


8-732S918 


•001501502 


667 


444899 


296740963 


25-8263431 


8-7372604 


•001499250 


668 


446224 


298077632 


25-8456960 


8-7416246 


•001497006 


669 


447561 


299418309 


25-86?0343 


8-7459846 


•001494768 


670 


448900 


300763000 


25-8843582 


8-7503401 


•001492537 


671 


450241 


302111711 


25-9036677 


8-7546913 


•001490313 


672 


451584 


303464448 


25-9229628 


8-7590383 


•00148S095 


673 


452929 


304821217 


25-9422435 


8-7633809 


•001485884 


674 


454276 


306182024 


25-9615100 


8-7677192 


•001483680 


675 


455625 


307546875 


25-9807621 


8-7720532 


•001481481 


676 


456970 


308915776 


26-0000000 


S-7763830 


•001479290 



35 


Table of Squares 


, Cubes, Square 


and Cube Roots. 


Number. 


Squares. 


Cubes. 


\f PwOOtS. 


\/Roors. 


Reciprocals. 


677 


458329 


3102S8733 


26-0192237 


8-7807084 


•001477105 


678 


459684 


311665752 


26-0384331 


8-7850296 


•0C1474926 


679 


461041 


313046839 


26-0576284 


8-7893466 


•001472754 


680 


462400 


314432000 


26-0768096 


8-7936593 


•001470588 


681 


463761 


315821241 


26-0959767 


8-7979679 


•001468429 


682 


465124 


317214568 


26-1151297 


8-8022721 


•001466276 


683 


466489 


318611987 


26-1342687 


8-8065722 


•001464129 


6S4 


467856 


320013504 


26-1533937 


8-S10S6Sr 


•001461988 


685 


469225 


321419125 


26-1725047 


8-8151598 


•001459854 


686 


470596 


322828856 


26-1916017 


8-8194474 


•001457726 


6S7 


471969 


324242703 


26-2106848 


8-8237307 


•001455604 


6SS 


473344 


325660672 


26-2297541 


8-8280099 


•001453488 


689 


474721 


327082769 


26-2488095 


8-8322850 


•001451379 


690 


476100 


32S509000 


26-2678511 


8-8365559 


•001449275 


691 


477481 


329939371 


26-2868789 


8-8408227 


•001447178 


692 


478864 


331373888 


26-3058929 


8-8450854 


•001445087 


693 


480249 


332812557 


26-3248932 


8-8493440 


•001443001 


694 


481636 


334255384 


26-3438797 


8-8535985 


•001440922 


695 


483025 


335702375 


26-3628527 


8-8578489 


•001438849 


. 696 


484416 


337153536 


26-3S18119 


8-8620952 


•001436782 


697 


485809 


33860S873 


26-4007576 


8-8663375 


•001434720 


698 


487204 


340068392 


26-4196896 


8-8705757 


•001432665 


699 


48S601 


341532099 


26-43S6081 


8-S748099 


•001430615 


700 


490000 


343000000 


26-4575131 


8-S790400 


•001428571 


701 


491401 


344472101 


26-4764046 


8-SS32661 


•001426534 


702 


492804 


345948408 


26-4952S26 


8-8874882 


•001424501 


703 


494209 


347428927 


26-5141472 


8-S917063 


•001422475 


704 


495616 


348913664 


26-5329983 


8-8959204 


•001420455 


705 


497025 


350402625 


26-5518361 


8-9001304 


•001418440 


706 


498436 


351895816 


26-5706605 


8-9043366 


•001416431 


707 


499849 


353393243 


26-5894716 


8-9085387 


•001414427 


708 


501264 


354894912 


26-6082694 


8-9127369 


•001412429 


709 


502681 


356400829 


26-6270539 


8-9169311 


•001410437 


710 


504100 


357911000 


26-6458252 


8-9211214 


•001408451 


711 


505521 


359425131 


26-6645833 


8-9253078 


•001406470 


712 


506944 


360944128 


26-6833281 


8-9294902 


•001404494 


713 


508369 


362467097 


26-7020598 


8-9336687 


•001402525 


714 


509796 


363994344 


26-7207784 


8-937S433 


•001400560 


715 


511225 


365525875 


26-7394839 


8-9420140 


•001398601 


716 


512656 


367061696 


26-7581763 


8-9461809 


•001396648 


717 


514089 


36S601813 


26-7768557 


8-9503438 


•001394700 


718 


515524 


370146232 


26-7955220 


8-9545029 


•001392758 


719 


516961 


371694959 


26-8141754 


8-9586581 


•001390821 


720 


518400 


373248000 


26-^28157 


* 8-9628095 


•001388889 


721 


519841 


374805361 


26-8514432 


8-9669570 


•001386963 


722 


521284 


376367048 


26-8700577 


8-9711007 


•001385042 


723 


522729 


377933067 


26-88S6593 


8-9752406 


•0013S3126 


724 


524176 


379503424 


26-9072481 


8-9793766 


•001381215 


725 


525625 


381078125 


26-9258240 


8-9835089 


•001379310 


726 


527076 


382657176 


26-9443872 


8-9876373 


•001377410 


727 


528529 


3S42405S3 


26-9629375 


S-9917620 


•001375516 


728 


529984 


385828352 


•26-9814751 


8-9958899 


•001373626 



Table of Squares, Cubes, Square and Cube Roots. 



57 



Number. 


Squares. 


Cubes. 


V Roots. 


y/ Roots. 


Reciprocals. 


729 


531441 


387420489 


27-0000000 


9-0000000 


•001371742 


730 


532900 


389017000 


27-0185122 


9-0041134 


•001369863 


731 


534361 


390617891 


27-0370117 


9-0082229 


•001367989 


732 


535824 


392223168 


27-0554985 


9-0123288 


r001366120 


733 


537289 


393832837 


27-0739727 


9-0164309 


•001364256 


734 


538756 


395446904 


27-0924344 


9-0205293 


•001362398 


735 


540225 


397065375 


27-1108834 


9-0246239 


•001360544 


736 


541696 


398688256 


27-1293199 


9-0287149 


•001358696 


737 


543169 


400315553 


27*1477149 


9-0328021 


•001356852 


738 


544644 


401947272 


27-1661554 


9-0368857 


•001355014 


739 


546121 


403583419 


27-1845544 


9-0409655 


•001353180 


740 


547600 


405224000 


27-2029140 


9-0450419 


•001351351 


741 


549801 


406869021 


27-2213152 


9-0491142 


•001349528 


742 


550564 


408518488 


27-2396769 


9-0531831 


•001347709 


743 


552049 


410172407 


27-2580263 


9-0572482 


•001345895 


744 


553536 


411830784 


27-2763634 


90613098 


•001344086 


745 


555025 


413493625 


27-2946881 


9-0653677 


•001342282 


746 


556516 


415160936 


27-3130006 


9-0694220 


•001340483 


747 


558009 


416832723 


27-3313007 


9-0734726 


•001338688 


748 


559504 


418508992 


27-3495887 


9-0775197 


•001336898 


749 


561001 


420189749 


27-3678644 


9-0815631 


•001335113 


750 


562500 


421875000 


27-3861279 


9-0856030 


•001333333 


751 


564001 


423564751 


27-4043792 


9-0896352 


•001331558 


752 


565504 


425259008 


27-4226184 


9-0936719 


,-001329787 


753 


567009 


426957777 


27-4408455 


9-0977010 


•001328021 


754 


568516 


428661064 


27-4590604 


9-1017265 


•001326260 


755 


570025 


430368875 


27-4772633 


9-1057485 


•001324503 


756 


571536 


432081216 


27-4954542 


9-1097669 


•001322751 


757 


573049 


433798093 


27-5136330 


9-1137818 


•001321004 


758 


574564 


435519512 


27-5317998 


9-1177931 


•001319261 


759 


576081 


437245479 


27-5499546 


9-1218010 


•001317523 


760 


577600 


488976000 


27-5680975 


9-1258053 


•001315789 


761 


579121 


440711081 


27-5862284 


9-1298061 


•001314060 


762 


580644 


442450728 


27-6043475 


9-1338034 


•001312336 


763 


582169 


444194947 


27-6224546 


9-1377971 


•001310616 


764 


583696 


445943744 


27-6405499 


9-1417874 


•001308901 


765 


585225 


447697125 


27-6586334 


9-1457742 


•001307190 


766 


586756 


449455096 


27-6767050 


9-1497576 


•001305483 


767 


588289 


451217663 


27*6947648 


9-1537375 


•001303781 


768 


589824 


452984832 


27-7128129 


9-1577139 


•001302083 


769 


591361 


454756609 


27-7308492 


9-1616869 


•001300390 


770 


592900 


456533000 


27-7488739 


9-1656565 


•001298701 


771 


594441 


458314011 


27-7668868 


9-1696225 


•001297017 


772 


595984 


46009^648 


27-7848880 


9-1735852 


•001295337 


773 


597529 


461889917 


27-8028775 


9-1775445 


•001293661 


774 


599076 ! 463684824 


27-8208555 


9-1815003 


•001291990 


775 


600625 465484375 


27-8388218 


9-1854527 


•001290323 


776 


602176 '467288576 


27-8567766 


9-1894018 


•001288660 


777 


603729,469097433 


27-8747197 


9-1933474 


•001287001 


778 


605284 j 470910952 


27-8926514 


9-1972897 


•001285347 


779 


606841 472729139 


27-9105715 


9-2012286 . 


•001283697 


780 


6084001474552000 


27-9284801 


9-2051641 


• -001282051 



SS ' 


Table of Squares, 


Cubes, Square 


axb Cube Roots. 


Number. 


Squares. Cubes. 




i^Roots. 


Reciprocals. 
•001280410 


V Roots. 


781 


609961 1 476379541 


27-9463772 


9-2090962 


782 


611524 


1 47821176S 


27-9642629 


9-2130250 


• -001278772 


783 


6130S9 


! 480048687 


27-9821372 


9-2169505 


•001277139 


784 


614656 


: 4S1890304 


28-0000000 


9-220S726 


•001275510 


785 


616225 


i 483736625 


28-0178515 


9-2247914 


•001273SS5 


786 


617796 


485587656 


28-0356915 


9-2287068 


•001272265 


787 


619369 


487443403 


2S-0535203 


9-2326189 


•00127064S 


788 


620944 


4S9303872 


28-0713377 


9-2365277 


•001269036 


789 


622521 


491169069 


28-0S9143S 


9-2404333 


•001267427 


790 


624100 


493039000 


28-10693S6 


9-2443355 


•001265823 


791 


6256S1 


494913671 


28-1247222 


9-24S2344 


•001264223 


792 


627624 


496793088 


28-1424946 


9-2521300 


•001262626 


793 


628849 


498677257 


28-1602557 


9-2560f24 


•001261034 


794 


630436 


500566184 


28-1780056 


9-2599114 


•001259446 


795 


632025 


502459875 


28-1957444 


9-2637973 


•001257862 


796 


633616 


504358336 


28-2134720 


9-2676798 


•001256281 


797 


635209 


506261573 


28-2311884 


9-2715592 


•001254705 


798 


636804 


508109592 


28-24S893S 


9-2754352 


•001253133 


799 


638401 


510082399 


28-26658S1 


9-2793081 


•001251364 


800 


640000 


512000000 


28-2842712 


9-2S31777 


•001250000 


801 


641601 


513922401 


28-3019434 


9-2870444 


•001248439 


802 


643204 


515S49608 


2S-3196045 


9-2909072 


•00124GS83 


803 


644S09 


517781627 


28-3372546 


9-2947671 


•001245330 


804 


646416 


51971S464 


28-3548938 


9-2986239 


•001243781 


805 


648025 


521660125 


28-3725219 


9-3024775 


•001242236 


806 


649636 


523606616 


28-3901391 


9-3063278 


•001240695 


807 


651249 


525557943 


28-4077454 


9-3101750 


•001239157 


808 


652S64 


527514112 


28-4253408 


9-3140190 


•001237624 


809 


6544S1 


529475129 


28-4429253 


9-3178599 


•001236Q94 


810 


656100 


531441000 


28-4604959 


9-3216975 


•001234568 


811 


657721 


533411731 


28-4780617 


9-3255320 


•001233046 


812 


659344 


5353S7328 


28-4956137 


9-3293634 


•001231527 


813 


660969 


537367797 


28-5131549 


9-3331916 


•001230012 


814 


662596 


539353144 


28-5306852 


9-3370167 


•001228501 


815 


664225 


541343375 


28-5482048 


9-3408386 


•001226994 


816 


665856 


54333S496 


28-5657137 


9-3446575 


•001225499 


817 


667489 


54533S513 


28-5832119 


9-3484731 


•001223990 


818 


669124 


547343432 


28-6006993 


9-3522857 


•001222494 


819 


670761 


549353259 


28-6181760 


9-3560952 


•001221001 


820 


672400 


551368000 


28-6356421 


9-3599016 


•001219512 


821 


674041 


553387661 


2S-6530976 


9-3637049 


•001218027 


322 


G756S4 


555412248 


28-6705424 


9-3675051 


•001216545 


823 


677329 


557441767 


28-6879716 


9-3713022 


•001215067 


824 


678976 


559476224 


28-7054002 


Q,-3750963 


•001213592 


825 


6S0625 


561515625 


28-7223132 


9-3788873 


•001212121 


826 


682276 


563559976 


28-7402157 


9-3826752 


•001210654 


827 


683929 


565609283 


28-7576077 


9-3864600 


•001209190 


828 


685584 


567663552 


28-7749891 


9-3902419 


•001207729 


829 


687241 


569722789 


28-7923601 


9-3940206 


•001206273 


830 


688900 


5717S7000 


28-8097206 


9-3977964 


•001204819 


831 


690561 


573856191 


28-8270706 


9-4015691 


•001203369 


832 


692224 


575930368 


28-8444102 


9-4053387 ! 


•001201923 





Tabli 


of Squares 


Cubes, Square 


and Cube Roots. 39 


Number. 


Squares. 


Cubes. 


\/ Roots. 


V Roots. 


Reciprocals. 


833 


6938S9 


573009537 


28-8617394 


9-4091054 


•001200430 


834 


695556 


5S0093704 


28-8790532 


9-4123690 


•001199041 


835 


697225 


5S21S2875 


28-8963666 


9-4166297 


•001197605 


836 


698S96 


584277056 


28-9136646 


9-4203873 


•001196172 


837 


700569 


586376253 


28-9309523 


9-4241420 


•001194743 


833 


702244 


588480472 


28-9482297 


9-4278936 


•001193317 


839 


703921 


590589719 


23-9654967 


9-4316423 


•001191895 


840 


705600 


592704000 


28-9827535 


9-4353800 


•001190476 


841 


707281 


594823321 


29-0000000 


9-4391307 


•001139061 


842 


708964 


596947688 


29-0172363 


9-4428704 


•00118*7648 


843 


710649 


599077107 


29-0344623 


9-4466072 


•001186240 


844 


712336 601211584 


29-0516781 


9-4503410 


•001134834 


845 


714025! ef03351125 


29-0688837 


9-4540719 


•001183432 


846 


715716! 605495736 


29-0860791 


9-4577999 


•001182033 


847 


717409 607645423 


29-1032644 


9-4615249 


•001180638 


84S 


719104 


609800192 


29-1204396 


9-4652470 


•001179245 


849 


720801 


611960049 


29-1376046 


9-4689661 


•001177856 


850 


722500 


614125000 


29-1547595 


9-4726S24 


•001176471 


851 


724201 


616295051 


29-1719043 


9-4763957 


•0011750S8 


852 


725904 


618470208 


29-1890390 


9-4801061 


•001173709 


853 


727609 


620650477 


29-2061637 


9-4838136 


•001172333 


854 


729316 


622835864 


29-2232784 


9*4875182 


•001170960 


855 


7ai025 


625026375 


29-2403830 


9-4912200 


•001169591 


856 


732736 


627222016 


29-2574777 


9*4949188 


•001168224 


857 


734449 


629422793 


29-2745623 


9-4986147 


•001166861 


858 


736164 


631628712 


29-2916370 


9*5023078 


•001165501 


859 


737881 


633S39779 


29-30S7018 


9*5059980 


•001164144 


860 


739600 


636056000 


29-3257566 


9-5096S54 


•001162791 


861 


741321 


63S277381 


29-3428015 


9*5133699 


•001161440 


862 


743044 


640503928 


29-3598365 


9-5170515 


•001160093 


863 


744769 642735647 


29-376S616 


9*5207303 


•001158749 


864 


746496,644972544 


29-3938769 


9-5244063 


•001157407 


865 


74S225 J 647214625 


29-4108823 


9-5280794 


•001156069 


866 


749956 


649461896 


29-4278779 


9*5317497 


•001154734 


867 


751689 


651714363 


29-4448637 


9*5354172 


•001153403 


868 


753424 


653972032 


29-4618397 


9-5390818 


•001152074 


869 


755161 


656234909 


29-4788059 


9*5427437 


•001150748 


870 


756900i.658503000 


29-4957624 


9-5464027 


•001149425 


871 


758641 660776311 


29-5127091 


9-55005S9 


•001148106 


872 


760384|663054848 


29-5296461 


9-5537123 


•001146789 


873 


762129,665338617 


29-5465734 


9-5573630 


•001145475 


874 


763876 667627024 


29-5634910 


9-5610108 


•001144165 


875 


765625 669921875 


29-5S03989 


9-5646559 


•001142S57 


876 


767376 : 672221376 


29-5972972 


9-5682782 


•001141553 


877 


769129 1 674526133 


29-6141858 


9-5719377 


•001140251 


878 


770884 676836152 


29-6310848 


9-5755745 


•001138952 


879 


772641 ! 679151439 


29-6479342 


9-5792085 


•001137656 


8S0 


774400 681472000 


29-6647939 


9-5828397 


•001136364 


881 


776161 


683797841 


29-6816442 


9-53646S2 


•001135074 


882 


777924 


686128963 


29-6984848 


9-5900937 


•0011337S7 


8S3 


779689 


688465387 


29-7153159 


9-5937169 


•001132503 


884 


781456 


690807104 


29-7321375 


9-5973373 


•001131222 



40 


Table op Squares 


, Cubes, Square and Cube Roots. 


Number. 


Squares. 


Cubes. 






Reciprocals. 


y/ Roots. 


s/ Roots. 


885 


783225 


693154125 


29-7489496 


9-6009548 


•001129944 


886 


784996 695506456 


29-7657521 


9-6045696 


•001128668 


887 


786769 ! 697864103 


29-7825452 


9-6081817 


•001127396 


8S8 


788544 


700227072 


29-7993289 


9-6117911 


•001126126 


889 


790321 


702595369 


29-8161030 


9-6153977 


•001124859 


890 


792100 


704969000 


29-8328678 


9-6190017 


•001123596 


891 


793881 


707347971 


29-8496231 


9-6226030 


•001122334 


892 


795664 


707932288 


29-8663690 


9-6262016 


•001121076 


893 


797449 


712121957 


29-8831056 


9-6297975 


•001119821 


894 


799236 


714516984 


29-8998328 


9-6333907 


•001118568 


895 


801025 


716917375 


29-9165506 


9-6369812 


•001117818 


896 


802816 


719323136 


29-9332591 


9-6405690 


•001116071 


897 


804609 


721734273 


29-9499583 


9-6441542 


•001114827 


898 


806404 


724150792 


29-9666481 


9-6477367 


•001113586 


899 


808201 


726572699 


29-9833287 


9-6513166 • 


•001112347 


900 


810000 


729000000 


30-0000000 


9-6548938 


•001111111 


901 


811801 


731432701 


30-0166621 


9-6584684 


•001109878 


902 


813604 


733S70808 


30-0333148 


9-6620403 


•001108647 


903 


815409 736314327 


30-0499584 


9-&656096 


•001107420 


904 


817216 738763264 


30-0665928 


9-6691762 


•001106195 


905 


819025 1741217625 


30-0832179 


9-6727403 


•001104972 


906 


820836 743677416 


30-0998339 


9-6763017 


•001103753 


907 


822649 746142643 


30-1164407 


9-6798604 


•001102536 


908 


824464 748613312 


20-1330383 


9-6834166 


•001101322 


909 


826281 ,751089429 


30-1496269 


9-6869701 


•001100110 


910 


828100 753571000 


30-1662063 


9-6905211 


•001098901 


911 


829921 1756058031 


30-1827765 


9-6940694 


•001097695 


912 


831744 1758550825 


30-1993377 


9-6976151 


•001096491 


913 


833569 j 761048497 


30-2158899 


9-7011583 


•001095290 


914 


83.^396 {763551944 


30-2324329 


9-7046989 


•001094092 


915 


837225 


766060875 


30-2489669 


9-7082369 


•001092896 


916 


839056 


768575296 


30-2654919 


9-7117723 


•001091703 


917 


840889 


771095213 


30-2820079 


9-7153051 


•001090513 


918 


842724 


773620632 


30-2985148 


9-7188354 


•001089325 


919 


844561 776151559 


30-3150128 


9-7223631 


•001088139 


920 


846400 i 778688000 


30-3315018 


9-7258883 


•001086957 


921 


848241 | 781229961 


30-3479818 


9.7294109 


•001085776 


922 


850084 J783777448 


30-3644529 


9-7329309 


•001084599 


923 


851929 1786330467 


30-3S09151 


9-7364484 


•001083423 


924 


853776 788889024 


30-3973683 


9-7399634 


•001082251 


925 


855625 791453125 


30-4138127 


9-7434758 


•001081081 


926 


857476 


794022776 


30-4302481 


9-7469857 


•001079914 


927 


859329 


796597983 


30-4466747 


9-7504930 


•001078749 


928 


861184 


799178752 


30-4630924 


9-7539979 


•001077586 


929 


863041 


801765089 


30-4795013 


9-7575002 


•001076426 


930 


864900 


804357000 


30-4959014 


9-7610001 


•001075269 


931 


866761 


806954491 


30-5122926 


9-7644974 


•001074114 


932 


868624 


809557568 


30-5286750 


9-7679922 


•001072961 


933 


870489 812166237 


30-5450487 


9-7714845 


•001071811 


934 


872356 


814780504 


30-5614136 


9-7749743 


•001070664 


935 


874225 


817400375 


30-5777697 


9-7784616 


•001069519 


936 


876096 820025856 


30-5941171 


9-7829466 


•001068376 





Tabu 


of Squares, 


Cubes, Square 


and Cube Roots 


41 


Number. 


Squares. | Cubes. 




y/ Roots. 


Reciprocals. 


V Hoots. 


937 


877969 j 822656953. 


30-6104557 


9-7854288 


-001067236 


938 


879844 825293672 


30-6267857 


9-78S9087 


•0010660GS 


939 


881721 


827936019 


30-6431069 


9-7923861 


•001064963 


940 


883600 


830584000 


30-6594194 


9-7958611 


•001063830 


941 


885481 


833237621 


30-6757233 


9-7993336 


•001062699 


942 


887364 


835896888 


30-6920185 


9-8028036 


•001061571 


943 


889249 838561807 


30-7083051 


9-8062711 


•001060445 


944 


891136 841232384 


30-7245830 


9-8097362 


•001059322 


945 


893025 843908625 


30-7408523 


9-8131989 


•001058201 


948 


894916 


846590536 


30-7571130 


9-8166591 


•001057082 


947 


896808 


84927S123 


30-7733651 


9-8201169 


•001055966 


948 


898704 


851971392 


30-7896086 


9-8235723 


•001054852 


949 


900601 


854670349 


30-8058436 


9-8270252 


•001053741 


950 


902500 857375000 


30-8220700 


9-8304757 


•001052632 


951 


904401 


860085351 


30-8382879 


9-8339238 


•001051525 


952 


906304 


862801408 


30-8544972 


9-8373695 


•001050420 


953 


908209 


865523177 


30-8706981 


9-8408127 


•001049318 


954 


910116 


868250664 


30-8868904 


9-8442536 


•001048218 


955 


912025 


870983875 


30-9030743 


9-8476920 


•001047120 


956 


913936 


873722816 


30-9192477 


9-8511280 


•001046025 


957 


915849 


876467493 


30-9354166 


9-8545617 


•001044932 


958 


917764 


879217912 


30-9515751 


9-8579929 


•001043841 


959 


919681 


881974079 


30-9677251 


9-8614218 


•001042753 


960 


921600 


884736000 


30-9838668 


9-S6484S3 


•001041667 


961 


923521 


887503681 


31-0000000 


9-8682724 


•001040583 


962 


925444 


890277128 


31-0161248 


9-8716941 


•001039501 


963 


927369 


893056347 


31-0322413 


9-8751135 


•001038422 


964 


929296 


895841344 


31-04S3494 


•9-8785305 


•001037344 


965 


931225 


898632125 


31-0644491 


9-8819451 


•001036269 


9Q6 


933156 


901428696 


31-0805405 


9-8853574 


•001035197 


967 


935089 


904231063 


£1-0966236 


9-88S7673 


•001034126 


968 


937024 


907039232 


31-1126984 


9-8921749 


•001033058 


969 


938961 


909853209 


31-1287648 


9-8955801 


•001031992 


970 


940900 


912673000 


31-1448230 


9-8989830 


•001030928 


971 


942841 


915498611 


31-160S729 


9-9023835 


•001029866 


972 


944784 


918330048 


31-1769145 


9-9057817 


•001028807 


973 


946729 


921167317 


31-1929479 


9-9091776 


•001027749 


974 


948676 


924010424 


31-2089731 


9-9125712 


•001026694 


975 


950625 


926859375 


31-2249900 


9-9159624 


•001025641 


976 


952576 


929714176 


31-2409987 


9-9193513 


•001024590 


977 


954529 


932574833 


31-2569992 


9-9227379 


•001023541 


978 


956484 


935441352 


31-2729915 


9-9261222 


•001022495 


979 


958441 


938313739 


31-2889757 


9-9295042 


•001021450 


980 


960400 


941192000 


31-3049517 


9-9328839 


•001020408 


981 


962361 


944076141 


31-3209195 


9-9362613 


•001019168 


982 


964324 946966168 


31-3368792 


9-9396363 


•001018330 


993 


966289 949862087 


31-3528308 


9-9430092 


•001017294 


984 


968256 952763904 


31-3687743 


9-9463797 


•001016260 


985 


970225 955671625 


31-3847097 


9-9497479 


•001015228 


986 


972196 958585256 


31-4006369 


9-9531138 


•001014199 


987 


974169 961504803 


31-4165561 


9-9564775 


•001013171 


988 


976144.964430272 


31-4324673 


9-9598389 


•001012146 




4* 











42 


Table of Squares, 


Cubes, Square j 


im> Cube Rook 


!. 


Number. 


Squares. 


Cubes. 






Reciprocals. 


y/ Boots. 


f/ Roots. 


989 


978123 


967361669 


31-4483704 


9-9631981 


•001011122 


990 


980100 


970299000 


31-4642654 


9-9665549 


•001010101 


991 


982081 


973242271 


31-4801525 


9-9699055 


•001009082 


992 


984064 


976191488 


31-4960315 


9-9732619 


•001008065 


993 


986049 


979146657 


31-5119025 


9-9766120 


•001007049 


994 


988036 


982107784 


31-5277655 


9-9799599 


•001006036 


995 


990025 


985074875 


31*5436206 


9-9833055 


•001005025 


996 


992016 


988047936 


31-5594677 


9-9866488 


•001004016 


997 


994009 


991026973 


31-5753068 


9-9899900 


•001003009 


998 


996004 


994011992 


31-5911380 


9-9933289 


•001002004 


999 


998001 


997002999 


31-6069613 


9-9966656 


•001001001 


1000 


1000000 


1000000000 


31-6227766 


10-0000000 


•001000000 


1001 


1000201 


1003003001 


31*6385840 


10-0033222 


•0009990010 


3 002 


1004004 


1006012008 


31-6543866 


10-0066622 


•0009980040 


1003 


1006009 


1009027027 


31*6701752 


10-0099899 


•0009970090 


3 004 


1008016 


1012048064 


31-6859590 


100133155 


•0009960159 


1005 


1010025 


1015075125 


31-7017349 


10-0166389 


•0009950249 


1006 


1010036 


1018108216 


31-7175030 


10-0199601 


•0009940358 


1007 


1014049 


1021147343 


31-7332633 


10-0232791 


•0009930487 


1008 


1016064 


1024192512 


31-7490157 


10-0265958 


•0009920635 


1009 


1018081 


1027243729 


31*7647603 


10-0299104 


•0009910S03 


1010 


1020100 


1030301000 


31-7804972 


10-0332228 


•0009900990 


1011 


1020121 


1033364331 


31-7962262 


10-0365330 


•0009891197 


1012 


1024144 


1036433728 


31-8119474 


10-0398410 


•0009881423 


1013 


1026169 


1039509197 


31-8276609 


10-0431469 


•0009871668 


1014 


1028196 


1042590744 


31-8433666 


10-0464506 


•0009861933 


1015 


1030225 


1045678375 


31-8590646 


10-0497521 


•0009852217 


1016 


1032256 


1048772096 


31-8747549 


10-0530514 


•0009842520 


1017 


1034289 


1051871913 


31-8904374 


10-0563485 


•0009832842 


1018 


1036324 


1054977832 


31-9061123 


10-0596435 


•0009823183 


1019 


1038361 


1058089859 


31-9217794 


10-0629364 


•0009813543 


1020 


1040400 


1061208000 


31-9374388 


10-0662271 


•0009803922 


1021 


1042441 


1064332261 


31-9530906 


10-0695156 


•0009794319 


1022 


1044484 


1067462648 


31-9687347 


10-0728020 


•0009784736 


1023 


1046529 


1070599.167 


31-9843712 


10-0760863 


•0009775171 


1024 


1048576 


1073741824 


32-0000000 


10-0793684 


•0009765625 


1025 


1050625 


1076890625 


32-0156212 


10-0826484 


; 0009756098 


1026 


1052676 


1080045576 


32-0312348 


10-0859262 


•0009746589 


1027 


1054729 


1083206683 


32-0468407 


10-0892019 


•0009737098 


1028 


1056784 


1086373952 


32-0624391 


10-0924755 


•0009727626 


1029 


1058841 


1089547389 


32-0780298 


10-0957469 


•0009718173 


1030 


1060900 


1092727000 


32-0936131 


10-0990163 


•0009708738 


1031 


1062961 


1095912791 


32-1091887 


10-1022835 


•0009699321 


1032 


1065024 


1099104768 


32-1247568 


10-1055487 


•0009689922 


1033 


1067089 


1102302937 


32-1403173 


10-1088117 


•0009680542 


1034 


1069156 


1105507304 


32-1558704 


10-1120726 


•0009671180 


1035 


1071225 


1108717875 


32-1714159 


10-1153314 


•0009661836 


1036 


1073296 


1111934656 


32-1869539 


10-1185882 


•0009652510 


1037 


1075369 


1115157653 


32-2024844 


10-1218428 


•0009643202 


1038 


1077444 


1118386872 


32-2180074 


10-1250953 


•0009633911 


1039 


1079521 


1121622319 


32-2335229 


10-1283457 


•0009624639 j 


1040 


1081600 


1124864000 


32-2490310 


10-1315941 


•0009615385 





Table o? Squares, 


Cubes, Square and Cube Roots 


!. 43 


Number. 


Squares. 


Cubes. 




y Hoots. 


Reciprocals. 


sj Roots. 


1041 


1083681 


1128111921 


32-2645316 


10-1348403 


•0009606148 


1042 


1085764 


1131366088 


32-2800248 


10-1380845 


•0009596929 


1043 


1087849 


1134626507 


32-2955105 


10-1413266 


•0009587738 


1044 


1089936 


1137893184 


32-3109888 


10-1445667 


•0009578544 


1045 


1092025 


1141166125 


32-3264598 


10-1478047 


•0009569378 


1046 


1094116 


1144445336 


32-3419233 


10-1510406 


•0009560229 


1047 


1096209 


1147730823 


32-3573794 


10-1542744 


•0009551098 


1048 


1098304 


1151022592 


32-3728281 


10-1575062 


•0009541985 


1049 


1100401 


1154320649 


32-3882695 


10-1607359 


•0009532888 


1050 


1102500 


1157625000 


32-4037035 


10-1639636 


•0009523810 


1051 


1104601 


1160935651 


32-4191301 


10-1671893 


•0009514748 


1052 


1106704 


1164252608 


32-4345495 


10-1704129 


•0009505703 


1053 


1108809 


1167575877 


32-4499615 


10-1736344 


•0009496676 


1054 


1110916 


1170905464 


32-4653662 


10-1768539 


•9009487666 


1055 


1113125 


1174241355 


32-4S07635 


10-1S00714 


•0009478673 


1056 


1115136 


1177583616 


32-4961536 


10-1832868 


•0009469697 


1057 


1117249 


1180932193 


32-5115364 


10-1865002 


•0009460738 


1058 


1119364 


1184287112 


32-5269119 


10-1897116 


•0009451796 


1059 


1121481 


1187648379 


32-5422802 


10-1929209 


•0009442871 


1060 


1123600 


1191016000 


32-5576412 


10-1961283 


•0009433962 


1061 


1125721 


1194389981 


32-5729949 


10-1993336 


•0009425071 


1062 


1127844 


1197770328 


32-5883415 


10-2025369 


•0009416196 


1063 


1129969 


1201157047 


32-6035807 


10-2057382 


•0009407338 


1064 


1132096 


1204550144 


32-6190129 


10-2089375 


•0009398496 


1065 


1134225 


1207949625 


32-6343377 


10-2121347 


•0009389671 


1066 


1136356 


1211355496 


32-6496554 


10-2153300 


•0009380863 


1067 


11384S9 


1214707763 


32-6649659 


10-2185233 


•0009372071 


1068 


1140624 


1218186432 


32-6802693 


10-2217146 


•0009363296 


1069 


1142761 


1221611509 


32-6955654 


10-2249039 


•0009354537 


1070 


1144900 


1225043000 


32-7108544 


10-2280912 


•0009345794 


1071 


1147041 


1228480911 


32-7261363 


10-2312766 


•0009337068 


1072 


1149184 


1231925248 


32-7414111 


10-2344599 


•0009328358 


1073 


1151329 


1235376017 


32-7566787 


10-2376413 


•0009319664 


1074 


1153476 


1238833224 


32-7719392 


10-2408207 


•0009310987 


1075 


1155625 


1242296875 


32-7871926 


10-2439981 


•0009302326 


1076 


1157776 


1245766976 


32-8024398 


10-2471735 


•0009293680 


1077 


1159929 


1249243533 


32-8176782 


10-2503470 


•0009285051 


1078 


1162084 


1252726552 


32-8329103 


10-2535186 


•0009276438 


1079 


1164241 


1256216039 


32-8481354 


10-2566881 


•0009267841 


1080 


1166400 


1259712000 


32-8633535 


10-2598557 


'0009259259 


1081 


1168561 


1263214441 


32-8785644 


10-2630213 


•0009250694 


1082 


1170724 


1266723368 


32-8937684 


10-2661850 


•0009242144 


1083 


1172889 


1270238787 


32-9089653 


10-2693467 


•0009233610 


1084 


1175056 


1273760704 


32-9241553 


10-2725065 


•0009225092 


1085 


1177225 


1277289125 


32-9393382 


10-2756644 


•0009216590 


1086 


1179396 


1280824056 


32-9545141 


10-2788203 


•0009208103 


1087 


1181569 


1284365503 


32-9696830 


10-2819743 


•0009199632 


1088 


1183744 


1287913472 


32-9848450 


10-2851264 


•0009191176 


1089 


1185921 


1291467969 


33-0000000 


10-2882765 


•0009182736 


1090 


1188100 


1295029000 


33-0151480 


10-2914247 


•0009174312 


1091 


1190281 


1298596571 


33-0302891 


10-2945709 


•0009165903 


1092 


1192464 


1302170688 


33-0454233 


10-2977153 


•0009157509 



44 



Table op Squares, Cubes, Square ant> Cube Roots. 



Number. 
1093 
1094 
1095 
1096 
1097 
1098 
1099 
1100 
1101 
1102 
1103 
1104 
1105 
1106 
1107 
1108 
1109 
1110 

mi 

1112 
1113 
1114 
1115 

1116 
1117 
1118 
1119 
1120 
1121 
1122 
1123 
1124 
1125 
1126 
1127 
1128 
1129 
1130 
1131 
1132 
1133 
1134 
1135 
1136 
1137 
1138 
1139 
1140 
1141 
1142 
1143 
1144 



Squares. 

1194649 
1196836 
1199025 
1201216 
1203409 
1205604 
1207801 
1210000 
1212201 
1214404 
1216609 
1218816 
1221025 
1223236 
1225449 
1227664 
1229881 
1232100 
1234321 
1236544 
1238769 
1240996 
1243225 
1245456 
1247689 
1249924 
1252161 
1254400 
1256641 
1258884 
1261129 
1263376 
1265625 
1267876 
1270129 
1272384 
1274641 
1276900 
1279161 
1281424 
1283689 
1285956 
1288225 
1290496 
1292769 
1295044 
1297321 
1299600 
1301881 
1304164 
1306449 
1308736 



Cubes. 

1305751357 
1309338584 
1312932375 
1316532736 
1320139873 
1323753192 
1327373299 
1331000000 
1334633301 
133S273208 
1341919727 
1345572864 
1349232625 
1352899016 
1356572043" 
1360251712 
1363938029 
1367631000 
1371330631 
1375036928 
1378749897 
1382469544 
13S6195875 
1389928896 
1393668613 
1397415032 
1401168159 
1404928000 
1408694561 
1412467848 
1416247867 
1420034624 
1423828125 
1427628376 
1431435383 
1435249152 
1439069689 
1442897000 
1446731091 
1450571968 
1454419637 
1458274104 
1462135375 
1466003453 
1469878353 
1473760072 
1477648619 
1481544000 
1485446221 
1489355288 
1493271207 
1497193984 



oots. 



33*0605505 
33-0756708 
33-0907842 
33*1058907 
33-1209903 
33-1360830 
33-1511689 
33-1662479 
33-1813200 
33-1963853 
33-2114438 
33-2266955 
33-2415403 
33-2565783 
33*2716095 
33*2866339 
33*3016516 
33*3166625 
33*3316666 
33-346664(7 
33-3616546 
33-3766385 
33*3916157 
33-4085862 
33*4215499 
33*4365070 
33-4514573 
33*4664011 
33*4813381 
33-4962684 
33-5111921 
33-5261092 
33-5410196 
33*5559234 
33-5708206 
33-5857112 
33-6005952 
33-6154726 
33-6303434 
33-6452077 
33-6600653 
33-6749165 
33-6897610 
83*7045991 
33-7174306 
33-7340556 
33-7490741 
33-7638880 
33-7786915 
33-7934905 
33-8082830 
33-8230891 



-{/Boots. 

10-3008577 
10-3039982 
10-3071368 
10-3102735 
10-3134083 
10-3165411 
10-3196721 
10-322S012 
10-3259284 
10-3290537 
10-3321770 
10-3352985 
10-3384181 
10-3415358 
10-3446517 
10-3477657 
10-3508778 
10-3539880 
10-3570964 
10-3802029 
10-3633076 
10-3664103 
10-3695113 
10-3726103 
10-3757076 
10-3788030 
10-3818965 
10-3849882 
10*3880781 
10-3911661 
10-3942527 
10-3973366 
10-4004192 
10-4034999 
10-4065787 
10-4096557 
10-4127310 
10-4158044 
10-4188760 
10-4219458 
10-4250138 
10-4280800 
10-4311443 
10-4342069 
10-4372677 
10-4403677 
10-4433839 
10-4464393 
10-4494929 
10-4525448 
10-4555948 
10-4586431 



Reciprocals. 

•0009149131 
•0009140768 
•0009132420 
•0009124008 
•0009115770 
•0009107468 
•0009099181 
•0009090909 
•0009082652 
•0009074410 
•0009066183 
•0009057971 
•0009049774 
•0009041591 
•0009033424 
•0009025271 
•0009017133 
•0009009009 
•0009000900 
•0008992806 
•0008984726 
•0008976661 
•0008968610 
•0008960753 
•0008952551 
•0008944544 
•0008936550 
•0008928571 
•0008960607 
•0008912656 
•0008904720 
•0008898797 
•0008S88889 
•0008880995 
•0008873114 
•0008865248 
•0008857396 
•0008849558 
•0008841733 
•0008833922 
•0008826125 
•0008818342 
•0008810573 
•0008802817 
•0008795075 
•0008787348 
•0008779631 
•0008771930 
•0008/64242 
•0008756567 
•0008748906 
•0008741259 





Table of Squares, 


Cubes, Square 


and Cube Roots 


45 


Number, 


Squares. 


Cubes. 




4 Roots. 


Reciprocals. 


V Roots. 


1145 


1311025 


1501123625 


33*8378486 


10-4616896 


•0008733624 


1146 


1313316 


1505060136 


33-8526218 


10-4647343 


•0008726003 


1147 


1315609 


1509003523 


33*8673884 


10-4677773 


•0008718396 


1148 


1317904 


1512953792 


33-8821487 


10-4708158 


•0008710801 


1149 


1320201 


1516910949 


33-8969025 


10-4738579 


-0008703220 


1150 


1322500 


1520875000 


33*9116499 


10-4768955 


•0008695652 


1151 


1324801 


1524845951 


33*9263909 


10-4799314 


•0008688097 


1152 


1327104 


1528823808 


33*9411255 


10-4829656 


•0008680556 


1153 


1329409 


1532808577 


33-9558537 


10-4859980 


•0008673027 


1154 


1331716 


1536800264 


33*9705755 


10-4590286 


•0008665511 


1155 


1334025 


1540798875 


33*9852910 


10-4920575 


•0008658009 


1156 


1336336 


1544804416 


34*0000000 


10-4950847 


•0008650519 


1157 


1338649 


1548816893 


34*0147027 


10-4981101 


•0008643042 


1158 


1340964- 


1552836312 


34-0293990 


10-5011337 


•0008635579 


1159 


1343281 


1556862679 


34-0440890 


10-5041556 


•0008628128 


1160 


1345600 


1560896000 


34*0587727 


10-5071757 


•0008620690 


1161 


1347921 


1564936281 


34-0734501 


10-5101942 


•0008613244 


1162 


1350244 


1568983528 


34*0881211 


10-5132109 


•0008605852 


1163 


1352569 


1573037749 


34-0127858 


10-5162259 


•0008598452 


1164 


1354896 


1577098944 


34*1174442 


10-5192391 


•0008591065 


1165 


1357225 


1581167125 


34-1320963 


10-5222506 


•0008583691 


1166 


1359556 


1585242296 


34*1467422 


10-5252604 


•0008576329 


1167 


1361889 


1589324463 


34-1613817 


10-5282685 


•0008568980 


1168 


1364224 


1593413632 


34-1760150 


10-5312749 


•0008561644 


1169 


1366561 


1597509809 


34-1906420 


10-5342795 


•0008554320 


1170 


1368900 


1601613000 


34-2052627 


10-5372825 


•0008547009 


1171 


1371241 


1605723211 


34-2198773 


10-5402837 


•0008539710 


1172 


1373584 


1609840448 


34-2344855 


10-5432832 


•0008532423 


1173 


1375929 


1613964717 


34-2490875 


10-5462810 


•0008525149 


1174 


1378276 


1618096024 


34*2636834 


10-5492771 


•0008517888 


1175 


1380625 


1622234375 


34-2782730 


10-5522715 


•0008510638 


1176 


1382976 


1626379776 


34-2928564 


10-5552642 


•0008503401 


1177 


1385329 


1630532233 


34-3074336 


10-5582552 


•0008496177 


1178 


1387684 


1634691752 


34*3220046 


10-5612445 


•0008488964 


1179 


1390041 


1638858339 


34*3365694 


10-5642322 


•0008481764 


1180 


1392400 


1643032000 


34*3511281 


10-5672181 


•0008471576 


1181 


1394761 


1647212741 


34-3656805 


10-5702024 


•0008467401 


1182 


1397124 


1651400568 


34-3802268 


10-5731849 


•0008460237 


1183 


1399489 


1655595487 


34-3947670 


10-5761658 


•0008453085 


1184 


1401856 


1659797504 


34*4093011 


10-5791449 


•0008445946 


1185 


1404225 


1664006625 


34-4238289 


10-5821225 


•0008438819 


11S6 


1406596 


1668222856 


34-4383507 


10-5850983 


•0008431703 


1187 


1408969 


1672446203 


34-4528663 


10-5880725 


•0008424600 


1188 


1411344 


1676676672 


34-4673759 


10-5910450 


•0008417508 


1189 


1413721 


1680914629 


34-4818793 


10-5940158 


•0008410429 


1190 


1416100 


1685159000 


34-4963766 


10-5969850 


•0008403361 


1191 


1418481 


1689410871 


34-5108678 


10-5999525 


•0008396306 


1192 


1420864 


1693669888 


34-5253530 


10-6029184 


•0008389262 


1193 


1423249 


1697936057 


34-5398321 


10-6058826 


•0008382320 


1194 


1425636 


1702209384 


34-5543051 


10-6088451 


•0008375209 


1195 


1428025 


1706489875 


34-5687720 


10-6118060 


•0008368201 


1196 1 


1430416' 1710777536 


34-5832329 


10-6147652 1 


•0008361204 



46 



Table op Squares, Cubes, Square and Cube Roots. 



Number. 


Squares. Cubes. 


s/ Roots. 


\/ Roots. 


Reciprocals. 


1197 


1432809; 171 5072373 


34*5976879 


10-6177228 


•0008354219 


1198 


1435204 11719374392 


34*6121366 


10-6206788 


•0008347245 


1199 


1437601 1 1723683599 


34*6265794 


10-6236331 


•00083402S4 


1200 


1440000 1728000000 


34-6410162 


10-6265S57 


•000S333333 


1201 


1442401 1732323601 


34*6554469 


34-6295367 


•0008326395 


1202 


1444S04 1736654408 


34*6698716 


10-6324860 


•000831946S 


1203 


1447209; 1740992427 


34-6842904 


10-6354338 


•0008312552 


1204 


1449616:1745337664 


34-6987031 


10-6383799 


•0008305648 


1205 


1452025; 1749690125 


34*7131099 


10-6413244 


•000829S755 


1206 


1454436 1754049816 


34-7275107 


10-6442672 


•0008291874 


1207 


1456849 1758416743 


34*7419055 


10-6472085 


•0008285004 


1208 


1459264:1762790912 


34*7562944 


10-6501480 


•0008278146 


1209 


1461681.1767172329 


34*7706773 


10-6530S60 


•0008271299 


1210 


1464100 1771561000 


34*7850543 


10-6560223 


•000S264463 


1211 


1466521 1775956931 


34*7994253 


10-6589570 


•0008257638 


1212 


1468944 1780360128 


34*8137904 


10-6618902 


•0008250825 


1213 


1471369 1784770597 


34-S2S1495 


10-6648217 


•0008244023 


1214 


1473796 : 1789188344 


34*8425028 


10-6677516 


•0008237232 


1215 


1476225 1793613375 


34-S56S501 


10-6706799 


•000S230453 


1216 


1478656 


1798045696 


34*8711915 


10-6736066 


•0008223684 


1217 


1481089 


1802485313 


34*8855271 


10-6765317 


•0008216927 


1218 


1483524 


1806932232 


34*8998567 


10-6794552 


•0008210181 


1219 


1485961 


1S11386459 


34-9H1805 


10-6S23771 


•000S203445 


1220 


1488400 


1815848000 


34-92849S4 


10-6852973 


•000S196721 


1221 


1490841 


1820316S61 


34-942S104 


10-6882160 


•0008190008 


1222 


1493284 


1S24793048 


34*9571166 


10-6911331 


•0008183306 


1223 


1495729 


1829276567 


34*9714169 


10-6940486 


•000S176615 


1224 


1498176 


1833764247 


34-9857114 


10-6969625 


•000S169935 


1225 


1500625 


1838265625 


35-0000000 


10-699S748 


•0008163265 


1226 


1503276 


1S42771176 


35*0142828 


10-7027855 


'0008156607 


1227 


1505529 


18472840S3 


35-02S559S 


10-7056947 


•0008149959 


1228 


15079S4 


1S51S0^352 
1856331989 


35-0428309 


10-7086023 


•0008143322 


1229 


1510441 


35*0570963 


10-7115083 


•O0OS136696 


1230 


1512900 


1S60S67000 


35-0713553 


10-7144127 


•00081300S1 


1231 


1515361 


1865409391 


35*0856096 


10-7173155 


•0008123477 


1232 


1517824 


1869959168 


35*099S575 


10-7202168 


•0008116883 


1233 


1520289 


1S74516337 


35*1140997 


10-7231165 


•000S11030O 


1234 


1522756 


1S790S0904 


35*1283361 


10-7260146 


•0008103728 


1235 


1525225 


1883652875 


35*1425568 


10-7289112 


•0008097106 


1236 


1527696 


1888232256 


35*1567917 


10-7318062 


•0008090615 


1237 


1530169 


1892819053 


35-171010S 


10-7346997 


•00080S4074 


1238 


1532644 


1897413272 


35*1852242 


10-7375916 


•000S077544 


1239 


1535121 


1902014919 


35*1994318 


10-7404S19 


•000SO71O25 


1240 


1537600 


1906624000 


35*2136337 


10-7433707 


•000S064516 


1241 


1540081 


1911240521 


35*2278299 


10-7462579 


•000805S018 


1242 


1542564 


1915S644SS 


35*2420204 


10-7491436 


•000S051530 


1243 


1545049 


1920495907 


35*2562051 


10-7520277 


•000S045052 


1244 


1547536 


19251347S4 


35-2703842 


10-7549103 


•0008038585 


1245 


1550025' 


1929781125 


35*2845575 


10-7577913 


•000S032129 


1246 


1552521: 


1934434936 


35*2987252 


10-7606708 


•0008025632 


1247 


1555009 : 


1939096223 


35*3128872 


10-^63548S 


•000S019246^ 


1248 


1557504! 1943764992 


35-3270435 ' 10*7664252 


•000S012821 





Tabli 


of Squares. 


Cu3es, Square and Cude Rooig 


47 


Number. 


Squares. 


Cubes. 






Reciprocals. 


1 V^Roots. 


%f Roots. 


1249 


1560001 


1948441249 


35-3411941 


10-7693001 


•0008006405 


1250 


1562500 


1953125000 


35-3553391 


10-7721735 


•0008000000 


1251 


1565001 


1957816251 


35-3694784 


10-7750453 


•0007993605 


1252 


1567504 


1962515003 


35-3836120 


10-7779156 ; 


•0007987220 


1253 


1570009 


1967221277 


35-3977400 


10-7307843 \ 


•0007930846 


1254 


1572516 


1971935064 


35-4118624 


10-7836516 i 


•0007974482 


1255 


1575025 


1976656375 


35-4259792 


10-7S65173 


•000796S127 


1256 


1577536 


1981385216 


35-4400903 


10-7893315 


•0007961783 


1257 


1580049 


1936121593 


35-4541958 


10-7922441 


•0007955449 


1258 


1582564 


1990365512 


35*4682957 


10-7951053 


•0007949126 


1259 


15S50S1 


1995616979 


35*4823900 


10-7979649 


•0007942812 


1260 


1587600 


2000376000 


35-4964787 


10-800S230 


•0007936508 


1261 


1590121 


2005142581 


35-5105618 


10-S036797 


•0007930214 


1262 


1592644 


2009916728 


35-5246393 


10-8065348 


•0007923930 


1263 


1595166 


201469S447 


35-5337113 


10-8093884 


•0007917656 


1264 


1597696 


2019487744 


35-5527777 


10-8122404 


•0007911392 


1265 


1600225 


2024234625 


35*5668385 


10-8150909 


•0007905138 


1266 


1602756 


2029089096 


35-5808937 


10-3179400 I 


•0007898894 


1267 


1605289 


2033901163 


35-5949434 


10-8207876 j 


•0007892660 


1268 


1607S24 


2038720S32 


35-6089376 


10-8236336 ! 


•00073S6435 


1269 


1610361 


2043548109 


35*6230262 


10-8264782 


•0007880221 


1270 


1612900 


2048383000 


35-6370593 


10-8293213 


•0007874016 


1271 


1615441 


2053225511 


35-6510S69 


10-8321629 


•0007367821 


1272 


16179S4 


205S075648" 


35*6651090 


10-3350030 


•0007861635 


1273 


1820529 


2062933417 


35-6791255 


10-8373416 


•0007855460 


1274 


1623076 


2067798S24 


35-6931366 


10-8406788 ! 


•0007349294 


1275 


1625625 


2072671875 


35-7071421 


10-8435144 


•0007843137 


1276 


1628176 


2077.552576 


35-7211422 


10-84634S5 


•0007836991 


1277 


1630729 


2082440933 


35-7351367 


10-8491812 J 


•0007S30S54 


127S 


16332S4 


2037336952 


35*7491258 


10-8520125 I 


•0007824726 


1279 


1635S41 


2092240639 


35*7631095 


10-S54S422 | 


•0007S1S608 


1280 


163S400 


2097152000 


35-7770876 


10-3576704 1 


•0007812500 


1231 


1640961 


2102071341 


35-7910603 


10-3604972 


•0007806401 


1282 


1643524 


2106997768 


35-8050278 


10-8633225 


•0007800312 


1283 


1646089 


2111932187 


35-8189894 


10-8661454 


•0007794232 


1284 


1648656 


2116874304 


35*8329457 


10-8689687 


•0007788162 


12S5 


1651225 


2121824125 


35-8468966 


10-8717897 


•00077S2101 


1286 


1653796 


2126781656 


35-860S421 


10-S746091 


•0007776050 


1287 


1656369 


2131746903 


35-3747822 


10-8774271 


•0007770008 


128S 


1653944 


2136719S72 


35-8S87169 


10-S802436 


•0007763975 


1289 


1661521 


2141700569 


35-9026461 


10-8830587 


•0007757952 


1290 


1664100 


2146689000 


35*9165699 


10-8858723 


•000775193S 


1291 


16666S1 


2151685171 


35-93048S4 


10-8886845 


•0007745933 


1292 


1669264 


2156689088 


35-9444015 


10-8914952 


•0007739938 


1293 


1671S49 


2161700757 


35-9583092 


10-8943044 


•0007733952 


1294 


1674436 


2166720184 


35-9722115 


10-8971123 


•0007727975 


1295 


1677025 


2171747375 


35-98610S4 


10-8999186 1 


•0007722008 


1296 


1679616 2176782336 


36-0000000 


10-9027235 ! 


•0007716049 


1297 


1682209 


2181825073 ! 


36-0138862 


10-9055269 


•0007710100 


129S 


1^84804 


2186875592 


36-0277671 


10-9083290 


•0007704160 


1299 


1637401 


2191933899 


36-0416426 


10-9111296 


•0007698229 


1300 


1690000 1 21970000001 


36-0555128 


10-9139237 


•0007692308 



48 


Tabli 


; of Squares, < 


3itbes, Square 


lnd Cube Roots. 


N timber. 








Reciprocals. 


Squares. 


Cubes. 


\f Roots. 


%/ Roots. 


1301 


1692601 


2202073901 


36-0693776 


10-9167265 


•0007686395 


1302 


1695204 


2207155608 


36-0832371 


10-9195228 


•0007680492 


1303 


1697809 


2212245127 


36-0970913 


10-9223177 


•0007674579 


1304 


1700416 


2217342464 


36-1109402 


10-9251111 


•0007668712 


1305 


1703025 


2222447625 


36-1247837 


10-9279031 


•0007662835 


1306 


1705636 


2227560616 


36-1386220 


10-9306937 


•0007656968 


1307 


1708249 


2232681443 


36-1524550 


10*9334829 


•0007651109 


1308 


1710864 


2237810112 


36-1662826 


10-9362706 


•0007645260 


1309 


1713481 


2242946629 


36*1801050 


10*9390569 


•0007639419 


1310 


1716100 


2248091000 


36-1939221 


10-9418418 


•0007633588 


1311 


1718721 


2253243231 


36*2077340 


10-9446253 


•0007627765 


1312 


1721344 


2258403328 


36-2215406 


10*9475074 


•0007621951 


1313 


1723969 


2263571297 


36*2353419 


10-9501880 


•0007616446 


1314 


1726596 


2268747144 


36-2491379 


10-9529673 


•0007610350 


1315 


1729225 


2273930875 


36-2626287 


10-9557451 


•0007604563 


1316 


1731856 


2279122496 


36-2767143 


10*9585215 


•0007598784 


1317 


17344S9 


2284322013 


36-2904246 


10-9612965 


•0007593014 


1318 


1737124 


2289529432 


36*3042697 


10-9640701 


•0007587253 


1319 


1739761 


2294744759 


36-3180396 


10-9668423 


•0007581501 


1320 


1742400 


2299968000 


36*3318042 


10-9696131 


•0007575758 


1321 


1745041 


2305199161 


36-3455637 


10*9723825 


•0007570023 


1322 


1747684 


2310438248 


36*3593179 


10-9751505 


•0007564297 


1323 


1750329 


2315685267 


36*3730670 


10-9779171 


•0007558579 


1324 


1752976 


2320940224 


36-3868108 


10-9806823 


•0007552870 


1325 


1755625 


2326203125 


36*4005494 


10-9834462 


•0007547170 


1326 


1758276 


2331473976 


36-4142829 


10-9862086 


•0007541478 


1327 


1760929 


2336752783 


36-4280112 


10*9889696 


•0007535795 


1328 


1763584 


2342039552 


36-4417343 


10*9917293 


•0007530120 


1329 


1766241 


2347334289 


36-4554523 


10-9914876 


•0007524454 


1330 


1768900 


2352637000 


36-4691650 


10*9972445 


•0007518797 


1331 


1771561 


2357947691 


36-4828727 


11-0000000 


•0007513148 


1332 


1774224 


2363266368 


36-4965752 


11-0027541 


•0007507508 


1333 


1776889 


2368593037 


36-5102725 


11-0055069 


•0007501875 


1334 


1779556 


2373927704 


36-5239647 


11-0082583 


•0007496252 


1335 


1782225 


2379270375 


36-5376518 


11-0110082 


•0007490637 


1336 


1784896 


2384621056 


36-5513388 


11-0137569 


•0007485030 


1337 


1787569 


2389979753 


36-5650106 


11-0165041 


•0007479432 


1338 


1790244 


2395346472 


36*5786823 


11-0192500 


•0007473842 


1339 


1792921 


2400721219 


36-5923489 


11-0219945 


•0007468260 


1340 


1795600 


2406104000 


36*6060104 


11-0247377 


•0007462687 


1341 


1798281 


2411494821 


36-6196668 


11-0274795 


•0007457122 


1342 


1800964 


2416893688 


36-6333181 


11-0302199 


•0007451565 


1343 


1803649 


2422300607 


36-6469144 


11-0329590 


•0007446016 


1344 


1806336 


2427715584 


36-6606056 


11-0356967 


•0007440476' 


1345 


1809025 


2133138625 


36-6742416 


11-0384330 


•0007434944 


1346 


1811716 


2438569736 


36-6878726 


11-0411680 


•0007429421 


1347 


1814409 


2444008923 


36-7014986 


11-0439017 


•0007423905 


1348 


1817104 


2449456192 


36-7151195 


11-0466339 


•0007418398 


1349 


1819801 


2454911549 


36-7287353 


11-0493649 


•0007412898 


1350 


1822500 


2460375000 


36-7423461 


11-0520945 


•0007407407 


1351 


1825201 


2465846551 


36-7559519 


11-0548227 


•0007401924 


1352 


1827904 


2471326208 


36-7695526 1 


11-0575497 1 


•0007396450 





Table of Squares, 


Cubes, Square and Cube Roots. 


49 


Number. 








Reciprocals. 


Squares. 


| Cubes. 


V^ Roots. 


\/ Roots. 


1353 


1830609 


2476813977 


36-7831483 


11-0602752 


•0007390983 


1354 


1833316 


2482309864 


36-7967390 


11-0629994 


•0007385524 


1355 


1836025 


2487813875 


36-8103246 


11-0657222 


•0007330074 


1356 


1838736 


2493326016 


36-8239053 


11-0684437 


•0007374631 


1357 


1841449 


2498846293 


36*8374809 


11-0711639 


•0007369197 


1358 


1844164 


2504374712 


36-8510515 


11-0738828 


•0007363770 


1359 


1846SS1 


2509911279 


36-8646172 


11-0766003 


•0007358352 


1360 


1849600 


2515456000 


36-8781778 


11-0793165 


•0007352941 


1361 


1852321 


2521008881 


36-8917335 


11-0820314 


•0007347539 


1362 


1855044 


2526569928 


36-9052842 


11-0847449 


•0007342144 


1363 


1857769 


£532139147 


36-9188299 


11-0874571 


•0007336757 


1364 


1860496 


2537716544 


36-9323706 


11-0901679 


•0007331378 


1365 


1863225 


2543302125 


36-9459064 


11-092S775 


•0007326007 


1366 


1865956 


2548895896 


36-9594372 


11-0955857 


•0007320644 


1367 


186S689 


2554497863 


36-9729631 


11-09S2926 


•0007315289 


1363 


1871424 


2560108032 


36-9864840 


11-1009982 


•0007309942 


1369 


1874161 


2565726409 


37-0000000 


11-1037025 


•0007304602 


1370 


1876900 


2571353000 


37*0135110 


11-1064054 


•0007299270 


1371 


1879641 


2576937811 


37-0270172 


11-1091070 


•0007293946 


1372 


18823S4 


2582630848 


37-0405184 


11-1118073 


•0007288630 


1373 


1885129 


2588282117 


37-0540146 


11-1145064 


•0007283321 


1374 


1887876 


2593941624 


37-0675060 


11-1172041 


•0007278020 


1375 


1890625 


2599609375 


37-0899924 


11-1199004 


•0007272727 


1376 


1893376 


26052S5376 


37-0944740 


11-1225955 


•0007267442 


1377 


1896129 


2610969633 


37-1079506 


11-1252893 


•0007262164 


1378 


1898884 


2616662152 


37*1214224 


11-1279817 


•0007256894 


1379 


1901641 


2622362939 


37-1348893 


11-1306729 


•0007251632 


1380 


1904400 


2623072000 


37-1483512 


11-1333628 


•0007246377 


1381 


1907161 


2633789341 


37-1618084 


11-1360514 


•0007241130 


1382 


1909924 


2639514968 


37-1752606 


11-1337386 


•0007235890 


1383 


1912639 


2645248887 


37-1887079 


11-1414246 


•0007230658 


1384 


1915456 


2650991104 


37-2021505 


11-1441093 


•0007225434 


1385 


1918225 


2656741625 


37-2155881 


11-1467926 


•0007220217 


1386 


1920996 


2662500456 


37*2290209 


11-1494747 


•0007215007 


1387 


1923769 


2668267603 


37-2424489 


11-1521555 


•0007209805 


1388 


1926544 


2674043072 


37-2558720 


11-1548350 


•0007204611 


1389 


1929321 


2679S26869 


37-2692903 


11-1575133 


•0007199424 


1390 


1932100 


2635619000 


37*2327037 


11-1601903 


•0007194245 


1391 


1934881 


2691419471 


37-2961124 


11-1628659 


•00071S9073 


1392 


1937664 


2697223283 


37*3095162 


11-1655403 


•0007183908 


1393 


1940449 


2703045457 


37-3229152 


11-1682134 


•0007178751 


1394 


1943236 


2708870984 


37-3363094 


11-1708852 


•0007173601 


1395 


1946025 


2714704875 


37-3496988 


11-1735558 


•0007168459 


1396 


1948816 2720547136 


37-3630834 


11-1762250 


•0007163324 


1397 


1951609 2726397773 


37-3764632 


11-1788930 


•0007158196 


1398 


1954404 


2732256792 


37*3898382 


11-1815598 


•0007153076 


1399 


1957201 


2738124199 


37-4032034 


11-1842252 


•0007147963 


1400 


1960000 


2744000000 


37-4165733 


11-186SS94 


•0007142857 


1401 


1962301 


2749884201 


37*4299345 


11-1895523 


•0007137759 


1402 


1965604 


2755776808 


37-4432904 


11-1922139 


•0007132668 


1403 


1968409 


2761677827 


37-4566416 


11-1948743 


•0007127584 


1404 


1971216 


2767587264 


37-4699880 


11-1975334 1 


•0007122507 



50 



Table of Squares, Cubes, Squabs and Cube Roots. 



Number. 


! Squares. 


Cubes. 




1405 


1974025 


2773505123 




1406 


1976836 


j 2779431416 




1407 


1979649 


2785366143 




1403 


19S2464 


2791309312 




1409 


1985281 


! 2797260929 




1410 


1988100 


2803221000 




1411 


1990921 


2809189531 




1412 


1993744 


2815166528 




1413 


1996569 


2S21151997 




1414 


1999396 


2827145944 




1415 


2002225 


2S33148375 




1416 


2005056 


2839159296 




1417 


2007889 


2845178713 




1418 


2010724 


2S51206632 




1419 


2013561 


2S57243059 


1420 


2016400 


2S63288000 


1421 


2019241 


2S69341461 




1422 


20220S4 


2875403448 




1423 


2024929 


2S81473967 




1424 


2027776 


2887553024 




1425 


2030625 


2S93640625 




1426 


2033476 


2899736776 




1427 


2036329 


2905S414S3 




1428 


2039184 


2911954752 




1429 


2042041 


291S076589 




1430 


2044900 


2924207000 




1431 


2047761 


2930345991 




1432 


2050624 


2936493568 




1433 


2053489 


2942649737 




1434 


2056356 


2948S14504 




1435 


2059225 


2954987875 




1436 


2062096 


2961169856 




1437 


2064969 


2967360453 




1438 


2067844 


2973559672 




1439 


2070721 2979767519 




1440 


2073600 29S5984000 




1441 


2076481 


2992209121 




1442 


2079364 


3098442888 




1443 


2082249 


3004685307 




1444 


2085136 


3010936384 




1445 


2088025 


3017196125 




1446 


2080916 


3023464536 




1447 


2093809 


3029741623 




1448 


2096704 


3036027392 




1449 


2099601 


3042321849 




1450 


2102500 


3048625000 




1451 


2105401 


3054936851 




1452 


2108304 


3061257408 




1453 


2111209 


3067586777 




1454 


2114116 


3073924664 




1455 


2117025 


3080271375 




1456 


2119936 


3086626816 





-y/ JtlOOtS. 

37-4833295 
37-4966665 
37-5099987 
37-5233261 
37-5366487 
37-5499667 
37-5632799 
37-5765SS5 
37-5898922 
37-6031913 
37-6164857 
37-6297754 
37-6430604 
37-6563407 
37-6696164 
37-6828874 
37-6961536 
37-7094153 
37-7226722 
37-7359245 
37-7491722 
37-7624152 
37-7756535 
37*7888873 
37-8021163 
37-8153403 
37-82S5606 
37-S417759 
37-8549864 
37-8631924 
37-881393S 
37-8945906 
37-9077828 
37-9209704 
37-934153S 
37-9473319 
37-9605058 
37-9736751 
37-986S398 
38-0000000 
38-0131556 
3S-0263067 
33-0394532 
3S-0525952 
38-0657326 
33-0788655 
33-0919939 
38-1051178 
3S-1182371 
38-1313519 
38-1444622 
38-1575681 



</ Hoots. 

11-2001913 
11-2028479 
11-2055032 
11-2081573 
11-2108101 
11-2134617 
11-2161120 
11-2187611 
11-22140S9 
11-2240054 
11-2267007 
11-2293443 
11-2319S76 
11-2346292 
11-2372696 
11-2399037 
11-2425465 
11-2451831 
11-2478185 
11-2504527 
11-2530856 
11-2557173 
11-2583478 
11-2609770 
11-2636050 
11-2662318 
11-26S8573 
11-2714316 
11-2741047 
11-2767266 
11-2793472 
11-2S19666 
11-2S45849 
11-2S72019 
11-2898177 
11-2924323 
11-2950457 
11-2976579 
11-3002683 
11-3028736 
11-3054871 
11-3080945 
11-3107006 
11-3133056 
11-3159094 
11-3185119 
11-3211132 
11-3237134 
11-3263124 
11-3289102 
11-3315067 
11-3341022 



Reciprocals. 

•0007117438 
•0007112376 
•0007107321 
•0007102273 
•0007097232 
•0007092199 
•0007087172 
•0007082153 
•0007077141 
•0007072136 
•0007067138 
•0007062147 
•0007057163 
•0007052186 
•0007047216 
•0007042254 
•0007037298 
•0007032349 
•0007027407 
•0007022472 
•0007017544 
•0007012623 
•0007007708 
•0007002301 
•0006997901 
•0006993007 
•00069SS120 
•0006983240 
•0006978367 
•0006973501 
•0006968641 
•000696378S 
•0006958942 
•0006954103 
•0006949270 
•0006944444 
•0006939625 
•0006934813 
•0006930007 
•0006925208 
•0006920415 
•0006915629 
•0006910350 
•0006906078 
•0006901312 
•0006896552 
•0006391799 
•0006387052 
•0006882312 
•0006877579 
•0006872852 
•0006868132 





Tabli 


of Squares, 


Cubes, Square and Cube Roots 


51 


Number. 








Reciprocals. 


Squares. 


Cubes. 


s/ Roots. 


ty Roots. 


1457 


2122849 


3092990993 


38-1706693 


11-3366984 


•0006863412 


1453 


2125764 


3099363912 


38-1837662 


11-3392894 


•0006858711 


1459 


212S6S1 


3.105745579 


38-1968585 


11-3418813 


•0006854010 


1460 


2131600 3112136000 


38-2099463 


11-3444719 


•0006849315 


1461 


2134521 


3118535181 


38-2230297 


11-3470614 


•0006S44627 


1462 


2137444 


3124943128 


38-2361085 


11-3496497 


•0006839945 


1463 


2140369 3131359847 


38-2491829 


11-3522368 


•0006835270 


1464 


2143296 3137785344 


38-2622529 


11-354S227 


•0006830601 


1465 


2146225 


3144219625 


38-2753184 


11-3574075 


•0006825939 


1466 


2149156 


3150662696 


38-2883794 


11-3599911 


•0006821282 


1467 


2152089 


3157114563 


38-3014360 


11-3625735 


•0006816633 


1468 


2155024 


3163575232 


3S-3144881 


11-3651547 


•0006811989 


1469 


2157961 


3170044709 


38-3275358 


11-3677347 


•0006807352 


1470 


2160900 


3176523000 


38-3405790 


11-3703138 


•0006802721 


1471 


2163841 


3183010111 


38-3536178 


11-372S914 


•0006798097 


1472 


2166784 


3189506048 


38-3666522 


11-3754679 


•0006793478 


1473 


2169729 


3196010817 


38-3796821 


11-3780433 


•0006788S66 


1474 


2172676 


3202524424 


3S-3927076 


11-3806175 


•0006784261 


1475 


2175625 


3209046875 


38-4057287 


11-3831906 


•0006779661 


1476 


2178576 


3215578176 


38-4187454 


11-3857625 


•0006775068 


1477 


2181529 


3222118333 


38-4317577 


11-3883332 


•0008770481 


1478 


2184484 


3228667352 


38-4447656 


11-3909028 


•0006765900 


1479 


2187441 


3235225239 


38-4577691 


11-3934712 


•0006761325 


1480 


2190400 


3241792000 


38-4707681 


11-3960384 


•0006756757 


1481 


2193361 


3248367641 


38-4837627 


11-3986045 


•0006752194 


14S2 


2196324 


3254952168 


38-4967530 


11-4011695 


•0006747638 


1483 


2199289 


3261545587 


38-5097390 


11-4037332 


•0006743088 


1484 


2202256 


3268147904 


38-5227206 


11-4062959 


•0006738544 


1485 


2205225 


3274759125 


38-5356977 


11-4088574 


•0006734007 


1486 


2208196 


3281379256 


38-5486705 


11-4114177 


•0006729474 


1487 


2211169 


3288008303 


38-5616389 


11-4139769 


'0006724950 


1488 


2214144 


3294646272 


38-5746030 


11-4165349 


'0006720430 


1489 


2217121 


3301293169 


38-5875627 


11-4190918 


'0006715917 


1490 


2220100 


3307949000 


38-6005181 


11-4206476 


•0006711409 


1491 


2223081 


3314613771 


38-6134691 


11-4242022 


•0006706908 


1492 


2226004 


3321287488 


38-6264158 


11-4267556 


•0006702413 


1493 


2229049 


3327970157 


38-6393582 


11-4293079 


'0006697924 


1494 


2232036 


3334661784 


38-6522962 


11-4318591 


'0006693440 


1495 


2235025 


3341362375 


38-6652299 


11-4344092 


'00066S8963 


1496 


223S016 


3348071936 


38-6781593 


11-43695S1 


•0006684492 


1497 


2241009 


3354790473 


38-6910843 


11-4395059 


•0006680027 


1498 


2244004 


3361517992 


38-7040050 


11-4420525 


•0006675567 


1499 


2247001 


3368254499 


38-7169214 


11-4445980 


•0006671114 


1500 


2250000 


3375000000 


38-7298335 


11-4471424 


•0006666667 


1501 


2253001 


3381754501 


38-7427412 


11-4496857 


•0006662225 


1502 


2256004 


33S8518003 


38-7556447 


11-4522278 


•0006657790 


1503 


2259009 3395290527 


38-7685439 


11-45476SS 


'0006553360 


1504 


2262016 3402072064 


38-7814389 


11-4573087 


•0006648936 


1505 


2265025 


340S862625 


38-7943294 


11-4598476 


•0006644518 


1506 


2268036 


34156622i6 


38-8072158 


11-4623850 


•0006640106 


1507 


2271049 


3422470843 


38-8200978 


11-4649215 


'0006835700 


1508 

1 


2274064 


3429288512 


38-8329757 11-4674568 


•0006631300 



52 



Table of Squares, Cubes, Square and Cube Roots. 



Number. 


Squares. 


Cubes. 


I 1 3/ 

y Roots. v Roots. 


Reciprocals. 


1509 


2277081 


3436115229 


38-8458491 


11-4699911 


•0006626905 


1510 


22S0100 


3442951000 


3S-85S71S4 


11-4725242 


•0006622517 


1511 


2283121 


3449795831 


38-8715834 


11-4750562 


•0006618134 


1512 


2286144 


3456649728 


38-8844442 


11-4775871 


•0006613757 


1513 


2289169 


3463512697 


38-8973006 


11-4801169 


•00066093S5 


1514 


2292196 


3470384744 


38-9101529 


11-4826455 


•0006605020 


1515 


2295225 3477265875 


38-9230009 


11-4851731 


•0006600660 


1516 


2298256 


3484156096 


38-9358447 


11-4876995 


•0006596306 


1517 


2301289 


3491055413 


33-9486841 


11-4902249 


•0006591958 


1518 


2304324 


3597963832 


38-9615194 


11-4927491 


•0006587615 


1519 


2307361 


3504881359 


38-9743505 


11-4952722 


•0006583278 


1520 


2310400 


3511808000 


38-9871774 


11-4977942 


•0006578947 


1521 


2313441 


3518743761 


39-0000000 


11-5003151 


•0006574622 


1522 


2316484 


3525688648 


39-0128184 


11-5028348 


•0006570302 


1523 


2319529 


3532642667 


39-0256326 


11-5053535 


•0006565988 


1524 


2322576 


3539605824 


39-0384426 


11-5078711 


•0006561680 


1525 


2325625 


3546578125 


39-0512483 


11-5103876 


•0006557377 


1526 


2328676 


3553559576 


39-0640499 


11-5129030 


•0006553080 


1527 


2331729 


3567549552 


39-0768473 


11-5154173 


•0006548788 


1528 


2334784 


3560558183 


39-0896406 


11-5179305 


•0006544503 


1529 


2337841 


3574558889 


39-1024296 


11-5204425 


•0006540222 


1530 


2340900 


3581577000 


39-1152144 


11-5229535 


•0006535948 


1531 


2343961 


3588604291 


39-1279951 


11-5254634 


•0006531679 


1532 


2347024 


3595640768 


39-1407716 


11-5279722 


•0006527415 


1533 


2350089 


3602686437 


39-1535439 


11-5304799 


•0006523157 


1534 


2353156 


3609741304 


39-1663120 


11-5329S65 


•0006518905 


1535 


2356225 


3616805375 


39-1790760 


11-5354920 


•0006514658 


1536 


2359256 


3623878656 


39-1918359 


11-5379965 


•0006510417 


1537 


2362369 


3630961153 


39-2045915 


11-5404998 


•0006506181 


1538 


2365444 


3638052S72 


39-2173431 


11-5430021 


•0006501951 


1539 


2368521 


3645153819 


39-2300905 


11-5455033 


•0006497726 


1540 


2371600 


3652264000 


39-2428337 


11-54S0034 


•0006493506 


1541 


2374681 


3657983421 


39-2555728 


11-5505025 


•0006489293 


1542 


2377764 


3666512088 


39-2683078 


11-5530004 


•0006485084 


1543 


2380849 


3673650007 


39-2810387 


11-5554972 


•0006480881 


1544 


2383936 


3680797184 


39-2937654 


11-5579931 


•0006476684 


1545 


2387025 


3687953625 


39-3064880 


11-5604878 


•0006472492 


1546 


2390116 


3695119336 


39-3192065 


11-5629815 


•0006468305 


1547 


2393209 


3702294323 


39-3319208 


11-5654740 


•0006464124 


1548 


2396304 


3709478592 


39-3446311 


11-5679655 


•0006459948 


1549 


2399401 


3716672149 


39-3573373 


11-5704559 


•0006455778 


1550 


2402500 


3723875000 


39-3700394 


11-5729453 


•0006451613 


1551 


2405601 


3731087151 


39-3827373 


11-5754336 


•0006447453 


1552 


2408704 


3738308608 


39-3954312 


11-5779208 


•0006443299 


1553 


2411809 


3745539377 


39-4081210 


11-5804069 


•0006439150 


1554 


2414916 


3752779464 


39-420S067 


11-5828919 


•0006435006 


1555 


2418025 


3760028875 


39-43348S3 


11-5853759 


•0006430868 


1556 


2421136 


3767287616 


39-4461658 


11-5878588 


•0006426735 


1557 


2424249 


3774555693 


39-4588393 


11-5903407 


•0006422608 


1558 


2427364 


3781833112 


39-4715087 


11-5928215 


•0006418485 


1559 


2430481 


3789119879 


39-4841740 


11-5953013 


•0006414368 


1560 


2433600 3796416000 


39-496S353 


11-5977799 


•0006410256 



Table op Squares, Cubes, Square and Cube Routs. 



63 



fumber 


Squares. 


1561 


2436721 


1502 


2439844 


1563 


2442969 


1564 


2446096 


1565 


2449225 


1566 


2452356 


1567 


2455489 


1563 


2458624 


1569 


2461761 


1570 


2464900 


1571 


2468041 


1572 


2471184 


1573 


2474329 


1574 


2477476 


1575 


2480625 


1576 


2483776 


1577 


2486929 


1578 


2490084 


1579 


2493241 


1580 


2496400 


1581 


2499561 


1582 


2502724 


1583 


2505889 


1584 


2509056 


1585 


2512225 


1586 


2515396 


1587 


2518569 


1588 


2521744 


1589 


2524921 


1590 


2528100 


1591 


2531281 


1592 


2534464 


1593 


2537649 


1594 


2540836 


1595 


2544025 


1596 


2547216 


1597 


2550409 


1598 


2553604 


1599 


2556801 


1600 


2560000 



Cubes. 

3803721481 
3811030328 
3818360547 
3825641444 
3833037125 
3840389496 
3847751263 
3855123432 
3862503009 
3869883000 
3877292411 
3884701248 
3892119157 
3S99547224 
3906984375 
3914430976 
3921887033 
3929352552 
3936S27539 
3944312000 
3951805941 
3959309368 
3906822287 
3974344704 
3981876825 
3989418056 
3996969003 
4004529472 
4012099469 
4014679000 
4027268071 
4034866688 
4042474857 
4050092584 
4057719875 
4065356736 
4073003173 
40S0659192 
4088324799 
4096000000 



\/ Roots. 

39-5094925 

39-5221457 

39-5347948 

39-5474399 

39-5600809 

39-5727179 

39-5853508 

39-5979797 

39-6106046 

39-6232255 

39-6358424 

39-6484552 

39-6610640 

39-6736688 

39-6862696 

39*6988665 

39-7114593 

39-7240481 

39-7366329 

39-7492138 

39-7617907 

39-7743636 

39-7869325 

39-7994976 

39-8120585 

39-8246155 

39-8371686 

39-S497177 

39-S622628 

39-8748040 

39-SS73413 

39-8998747 

39-9124041 

39-9249295 

39-9374511 

39-9499687 

39-9624824 

39-9749922 

39-9874980 

40-0000000 



fy Roots. 

11 6002576 
11-6027342 
11-6052097 
11-6076841 
11-6101575 
11-6126299 
11-6151012 
11-6175715 
11-6200407 
11-6225088 
11-6249759 
11-6274420 
11-6299070 
11-6323710 
11-6348339 
11-6372957 
11-6397566 
11-6422164 
11-6446751 
11-6471329 
11-6495895 
11-6520452 
11-6544998 
11-6569534 
11-6594059 
11 ; 6613574 
11-6643079 
11-6667574 
11-6692058 
11-6716532 
11-6740996 
11-6765449 
11-6789892 
11-6814325 
11-6838748 
11-6863161 
11-6887563 
11-6911955 
11-6936337 
11-6960709 



Reciprocals. 

0006406150 
0006402049 
0006397953 
0006393862 
0006389776 
0006385696 
0006381621 
0006377551 
0006373486 
0006369427 
0006365372 
0006361323 
0006357279 
0006353240 
0006349206 
0006345178 
0006341154 
0006337136 
0006333122 
0006329114 
0006325111 
0006321113 
0006317119 
0006313131 
0006309148 
0006305170 
0006301197 
0006297229 
0006293266 
0006289308 
0006285355 
0006281407 
0006277464 
0006273526 
0006269592 
0006265664 
0006261741 
0006257822 
0006253909 
0006250000 



To find the Square Root of Numbers exceeding 1600, 
Example 4. Require tbe Square Root of 34698. In the column of Squares 



you will find, 



+34969 = 187 3 , 
— 34698 = 18C 3 , + ... 

2U divided by 



-}-34969 = 1ST 3 , 

—34596 = 180 9 . 

373 = 000.727, 



^34698 = 186-727 nearly. 



54 Evolution. 



Wlien ifie number contains Integer and Decimals. 

Example 5. Required the Square Root of 784545 ? In the column of Squares 
you will find, 

+7849-96 = 88-62, +7849-96 = 88-62, 

—7845-45 = 88-52-, — 7832*2 5 = 88'52, 

451 divided by 1771 = 00-0256. 

^7845-45 = 88-5256 nearly. 

j&g^When the number of ciphers in the integer is even, the number of 
ciphers taken in the Square column must also be even ; but when the number 
of ciphers in the integer is odd, the number taken in the Square column must 
also be odd. 

To find the Cube Boot of Numbers exceeding 1600. 

Example 6. Required the Cube Root of 5694958 ? In the Cube column you will 
find, 

+5735339 = 1793 +5735339 = 1793. 

—5694958 = 1783- — 5639752 = 1783. 

40381 divided by 95587 = 000*4225, 

^5694958 = 178*4225 nearly. 

Wlien the number contains Integer and Decimals. 

Example 7. Required the Cube Root of 4186-586 ? In the column of Cubes you 
Will find, 

+4251-528 = 16-23 4251-528 = 16-23 

—4186-585 = 16-13- 4173-281 = 16-13 

54942 78247 = 00*083 

4' 4186-586 = 16*183 nearly. 

Jf^The following notice must be particularly attended to, when extracting 
Cube Hoot of numbers with decimals. 

2 ciphers in the integer must be 5, 8, or 11 ciphers in the Cube column. 



3, 6, or 9 

4, or 7 

5, or 8 

6, or 9 

7, or 10 



Example 8. Required the Cube Root of 61358*75 ? In the Cube column and 8 
ciphers you will find, 

+61629-875 = 3953 +61629875 = 39-53 

— 61358-750 = 3943- — 61162984 = 39-43 

271-125 divided by 466 891 = 00*05807 

^61358-75 = 39*45807. 

To find the Fourth Root, 

Rule. Extract the Square Root of the number as before described, and of 
that root extract the Square Root again, then the last is the Fourth root of 
the number. 

Example 9. Required the fourth root of 2469781 ? 



<§/ 2469781 = V ]/2469781 = yl571*4463 = 39*6467, the answer. 
To find tlie Sixth Root. 

Rule. Find the Cube Root of the number as before described, and of that 
root extract the_Cube Root again, and then the last is the Sixth root of the 
number. 



Nystrom's Calculator. 



NYSTROM'S CALCULATOR. 




All calculations in this Pocket Book have been computed "by this Instrument. 
It consists of a silvered brass plate on which are fixed two moveable arms, ex- 
tending from the centre to the periphery. On the plate are engraved a number 
of curved lines in such form and divisions, that by their intersection with the 
arms, numbers are read and problems solved. 

The arrangement for trigonometrical calculations is such that it is not neces- 
sary to notice sine, cosine, tan, &c, &c, operating only by the angles them- 
selves expressed in degrees and minutes. This makes trigonometrical solutions 
so easy, that any one who understands Simple Arithmetic, will be able to solvo 
trigonometrical questions. 

C Jcularions are performed by it almost instantly, no matter how complicated 
they may be, while there is nothing intricate or difficult in its use. The author 
of this book, who is the inventor, has thoroughly tested its practical utility. 
Without this instrument not one-tenth of the calculations and tables which 
he is continually bringing out, could be produced. 

ADVERTISEMENT. 

The attention of Engineers, Ship builders, and all whose business requires 
frequent and extensive calculations, is called to Nystrom's Ca'culator. Price, 
$20. To be obtained with description by applying to John W. Nystrom, Phila- 
delphia. 

Communications will be promptly attended to. 

This Calculator received the First Premium at the Franklin Institute 
Exhibition. 

Wm. J. YOUNO, 

Mathematical, Optical, and Calculating Machine Manufactwrer, 

4.3 N. 7th St., Philadelphia. 



» 



56 Logarithms. 



LOGARITHMS. 



A Logarithm is an exponent of a power to which 10 must he raised to give 
a certain number, which will be understood by this 



log. 100 = 2 because 102 = 100. 

log. 10000 =4 " 10* = 10000 

log. 5012 = 3-7 " 10 3 -7 = 5012. 

The unit of the logarithm is called the characteristic or index, and the decimal 
part is called the mantissa, the sum of the characteristic and mantissa is the Loga- 
rithm. The invariable number 10 is the base for the system of Loga- 
rithms. 

It is not necessary that the base should be 10, it can be any number, but nil 
the tables of logarithms now in common use, are calculated with 10 to the 
base. 

The nature of logarithms in connection with their numbers are such, that the 
index of the logarithm is always one less than the number of figures in the 
number, (when the base of the logarithm is 10,) as, 
index 5012 = 3 
mantissa 5012 = 0*7 
logarithm 5012 = 3-7T 
Let 10 be raised to any power x, and 

the power of 10* = a or log. a = a?, 
the power of 10* = b or log. b ~ z. 

Let the product of ab = c and the quotient- = c. 

b 

10*X10« = 10**2 = ab = c or log. c = x+ z. 
-— = 10*—* =— == d or log. d = x—z. 

10 a Q 

a = m* or log. m = zXlog. a. 

Vo== n or log. n = log. a : 3. 

Any number represented by the letters a, b, c, or d, can be a power of 10, which 
exponent is the logarithm for the number. Logarithms are calculated for every 
number with three figures in the accompanying Table, by which any operation in 
Multiplication, Division, Involution and Evolution can be performed by simple 
Addition or Subtraction of Logarithms. Tables of Logarithms are commonly more 
extensive, and calculated for any number of four or five figures, which would 
occupy too much room in this book ; but by the proportional parts, the logarithm 
can be found by this Table, to four or five figures. The index of the logarithms 
do not appear in the Table, only the mantissa. It is easily remembered that the 
index is one less, than the number of figures in t7ie number ; then when the num- 
ber is only one figure, the index is ; and when the number is a fraction, the 
index is negative. 

When the logarithm is to be found for a fraction, we commonly have the 
fraction expressed in a decimal ; and then the negative index is equal to the 
number of ciphers before the first figure, and commonly marked after the man- 
tissa; thus explained in whole numbers and fractions: 

log. 365 =2-56229.. log. 0-365 = -56229— 1. 

log. 46-7 = 1-66931 log. 0-0467 = -66931—2. 

log. 7-59 = 0*88024 log. 0-00759 = -88024—3. 

In the accompanying Table of Logarithms, for the trigonometrical lines the 

negative index is marked thus, 

log. sin. 35° 40' = log. 0-58306 = 1:76572. 



Logarithms/- 57 



To find the LogaritJim of Numbers. 
Example 1. Find the logarithm of 45 . 

To 45 in the first column of tho Tablf*. answers 65321 in the nest column, 
which is the mantissa ; index = 1 because 45 is two figures. 
Then, log. 45 = 1-85321, the answer 

Example 2. Find the logarithm of 768 ? 

Opposite 76 in the first column, answers 88536 in the column marked 8 on the 
top or bottom. Index = 2 because 768 is three figures. 
Then, log. 768 = 2-88536. 

Example 3. Find the Logarithm of 6846 1 

log. 6840 = 3-83505 
Proportional part, 64X0-6 = 384 

log. 6846 = 3-835434 the answer. 
To find the number for a given Logarithm* * 

Example, 1. "What number answers to the logarithm 3*87157 ? 
In the Table you will find in the column of logarithms, that 

log. 7440 = 3-87157. 
Example 2. What number answers to the logarithm 3-801884? 
Given logarithm 3-801884, 

Subt. nearest table log, 3-801400 = log. 6330, 
Divided by proportional part, 69|484| - - - - * - 7, 

6337 the req. numb. 
Multiplication by Logarithms. 
Rule. Add together the logarithms of the factors, and the sum is the loga- 
rithm of the product. 
Example 1. Multiply 425 by 48. 

To log. 425 = 2-62839, 

Add log. 48 = 1-68124 A 

The product, log. 20400 = 4*30963. 

Example 2. Multiply 79600 by 0-435. 

To log. 79600 = 4-90091, 

Add, log. 0-435 = -63848— 1, 

The product log. 34690 = 4-53939. 

Division by Logarithms. 
Rule. From the logarithm of the dividend subtract the logarithm of the di- 
visor, and the difference is the logarithm of the quotient. 

Example 1. Divide 43800 by 368. 

From log. 43800 = 4*64147, 

Subtract log. 368 = 2*56584, 

The quotient log, 119 = 2*07563. 

Example 2. Divide 36 by 0.625. 

From log. 36 = 1*55636, 

Subtract, log. 0-625 = '79588-1. 

The quotient, log. 57*6 = 1-76048. 

A negative index follows an opposite operation of its mantissa, as if the man- 
tissa is subtracted, add the negative index, and vice versa. 
Envolution by Logarithms. 
Rule. Multiply the logarithm of the number by its exponent, and the pro- 
duct is the logarithm of the power of the number. 
Involution by Logarithms. 
Rule. Divide the logarithm of the number by the index of the root, and the 
quotient is the logarithm of the root of the number. 



53 



LOGARITHM 07 NCMBSF.F 



LOGARITHM OF NUMBERS, FROM 
O to 1000, 





10 00000 

11 01139 

12 07918 

13 111394 

14 14612 

15 17609 

16 20412 

17 23044 
IS 25527 

19 27875 

20 ( 30103 

21 32221 

22 34242 

23 36172 

24 38021 

25 39794 



2Q 

27 
28 
29 
30 

31 
32 
33 
34 
35 
So 
37 
33 
39 
40 
41 
42 
48 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 



41497 
43136 
44715 
46239 
47712 
49136 
50515 
51551 
53147 
54406 
55630 
56820 
57978 
59106 
60206 
61278 
62324 
63346 
345 
65321 
66275 
67209 
68124 
69019 
69897- 
0757 
1600 
2427 
3239 
74036 



00000 
00432 
04532 
08278 
11727 
14921 
17897 
20682 
23299 
125767 
2S103 
30319 
32428 
34439 
36361 
38201 
39967 
41664 
43296 
44870 
46389 
47S56 
49276 
50650 
51982 
53275 
54530 
55750 
56937 
58092 
59217 
60314 
613S4 
62428 
63447 
64443 
65417 
66370 
67302 
68214 
69108 
69983 



2 j 3 

3010347712 

00860;01283 

04921J05307 

0863608990 

12057112385 

1522815533 

181S4;18469 

20951I2121S 

2355223804 

26007|26245 

28330 28555 

J30535 30749 30963 

32633 32838 33041 



60206 
01703 
05690 
09342 
12710 
15S36 
18752 
21484 
24054 
26481 
28780 



•^-,r 



34635 34830 35024 
36548 36735 36921 

383Sl!38560;38739 
l 40140|40312 ; 40483 
:41830l41995 42160 
434564361643775 

45024|4517S 45331 
46538 466S6 46834 
48000 4S144:4S237 



! 49415 49554J49693 
! 50785 l 50920'51054 
52113 : 52244 52374 
53402 5352153655 

!5465454777!54900 
'55S70 55990 56110 
57054 57170 57287 
5S206 58319 58433 
^932859439 59549 
6042260530 6063S 
6148961595 61700 
! 6253L62634 62736 
'63548,63648 03749 
64542 64640 647SS 
65513 65609 6570^ 
;66464!6655S|66651 
67394:67486 67577 
68304 68394 6S4S4 
69196 69284 69372 
70070 70156 ( 70243 
70842 70927 7101171096 
716S3 71767 71350 71933 
72509 7259172672 7275^ 
73319 73399 73480 73559 
74115 ,74193:74272 74351 



9 I>rop. 



69897,77815 84510 
02118,02530|02938 
06069 06445 06818 
09691:1003710380 



13672 
16731 
19590 
22271 



13033:13353 
1613616435 
1903319312 
21748:22010 
24303 24551124797 
26717(2695127184 
29003 29225'29446 
31175 3138631597 
33243'33445j33646 
35218 35410135602 
37106 37291|37474 
38916 39093 ! 39269 
40654 40S24J40993 
42324 42488 42651 



4424S 
:578S 
47275 
8713 
50105 
51454 
52763 



43933:44090 
4548445636 
46982J47129 
48430J48572 
4983149968 
5118851321 
52504 5203: 
53781 53907|54033 
55022 55145 55266 
5622956348,56466 
57403 5751857634 
5854658658 58771 
59659'59769p9379 
6074560S52 60959 
61804 61909,62013 
|62838 6294163042 
63848 63948 64048 
04S36'64933 65030 
65S0165S96J65991 
66745 I 66S3S,66931 
67669 67760 67851 
68574 68663 68752 
69460'6954S'69635 
70329|70415|70500 

71180;71265 ! 71349 
72015 7209872181 

7283572916:72997 
73639 73719.7379S 
74429 7450774585 



,90309 
03342 
^07188 
10721 
!l39S7 
117026 
J19S65 
J22530 
J25042 
27415 
129666 
|31806 
33845 
^35793 
! 37657 
39449 
i41162 
42813 
44404 
45939 
47421 
48855 
50242 
51587 
!52891 
J54157 
J5538S 
56584 
57749 
58883 
59988 
61066 

62117 
63144 
;64147 
|65127 
66086 
67024 
67942 
6SS42 
69722 
70586 
71433 
72263 
73078 
73878 
74663 



95424 
03742 
07554 
11059 
14301 
17318 
20139 
22788 
25285 
27646 
29885 
32014 
! 34044 
'35983 
37S39 
J39619 
41330 
j 42975 
44560 
46089 
47567 
4S995 

50379 
51719 
53020 
54282 
55509 
56702 
57863 
5S995 
60097 
61172 
62221 
63245 
64246 
65224 
661S1 
67117 
68033 
68930 
69S10 
70671 
71516 
72345 
73158 
73957 
74771 



415 

379 
34 
323 
300 
2S1 
264 
249 
236 
223 
212 
202 
194 
185 
177 
171 
164 
158 
153 
148 
143 
138 
134 
130 
126 
122 
119 
116 
113 
110 
107 
104 
102 
99 
98 
96 
94 
92 
90 



34 
^2 
31 
80 

Pjop 











LOGARITHMS OF NUMBERS 








69 


No. 


° 


1 


o 


3 


4 


5 


6 j 7 


8 


9 Prop. 


50 


74813 


74S96 


74973 


75050 


75127 


75204 


75281 7535S 


75434 


75511! 77 


57 


755S7 


75663 


75739 


75815 


75S91 


75966 


76042 


76117 


76192 


76267' 75 


58 


76342 


76417 


76492 


76566 


76641 


76715 


76789 


76863 


76937 


77011 


74 


5977085 


7715b 


77232 


77305 


77378 


77451 


77524 


77597 


77670 


77742 


73 


6077815 


77887 


77959 


78031 


78103 


7S175 


78247 


7S318 


78390 


78461 


72 


6178533 


78604 


78675 


7S746 


78816 


78887 


78958 


79023 


79098 


79169 


71 


62|79239 


79309 


79379 


7944S 


79518 


79588 


79657 


79726 


79796 


79865 


70 


6379934 


80002 


80071 


80140 


8020S 


30277 


80345 


30413 


80482 


30550 


69 


64 80618 


806S5 


80753 


80821 


80SSS 


S0956 


81023 


81090 


81157 


81224 


68 


65 ! S1291 


S135S 


81424 


S1491 


81557 


S1624 


81690 


81756 


81822 


S1S88 


67 


66 81954 


82020 


82085 


82151 


82216 


S22S2 


82347 


82412 


82477 


82542 


QQ 


67|S2607 


82672 


82736 


82801 


82866 


82930 


82994 


83058 


S3123 


83187 ^^ 


6883250 


S3314 


33378 


S3442 


83505 


S3569 


83632 


83695 


83758 


83821 64 


69 S3SS4 


83947 


S4010 


84073 


34136 


34198 


34260 


84323 


84385 


84447 


63 


7084509 


84571 


84633 


84695 


84751 


34818 


S48S0 


84941 


S5003 


85064 


62 


71 


S5125 


85187 


85248 


85309 


35369 


35430 


85491 


85551 


35612 


85672 


61 


72 


85733 


85793 


85853 


85913 


85973 


36033 


86093 


S6153 


S6213 


S6272 


60 


73 


S6332 


83991 


86451 


86510 


86569 


S6628 866S7 


S6746 


86S05 


86864 


59 


74 


S6923 


86981 


87040 


87093 


87157 


87215; 87273 


87332 


S7390 


8744S 


58 


75 


87506 


87564 


87621 


87679 


87737 


37794! 87S52 


87906 


S7966 


8S024 


57 


76:88081 


88138 


38195 


8S252 


88309 


38366 88422 


88479 


88536 


8S592 


56 


77:88649 


88705 


88761 


88818 


88874 


33930 


88986 


89 042 


39098 


89153 


56 


7889209 


89265 


S9320 


89376 


89431 


39487 


89542 


89597 


89652 


89707 


55 


7989762 


89817 


89872 


89927 


89982 


90036 


90091 


90145 


90200 


90254 


54 


8090309 


90363 


90417 


90471 


90525 


90579 90633 90687 


90741 


90794 


54 


8190848 


90902 


90955 


91009 


91062 


91115 


91169,91222 


91275 


91328 


53 


82191381 


91434 


91487 


91540 


91592 


91645 


91698 '91750 


91803 


91855 


53 


83:91907 


91960 


92012 


92064 


92116 


92108 


92220:92272 


92324 


92376 


52 


84| 9242 7 


92479 


92531 


92582 


92634 


926S5 


92737 '92788 


92839 


92S90 


51 


85'92941 


92993 


93044 


93095 


93176 


93196 


93247 


93293 


93378 


93399 


51 


86 93449 


93500 


93550 


93601 


93651 


93701 


93751 


93802 


93S52 


93902 


50 


8793951 


94001 


94051 


94101 


94151 


J4200 94250 


94300 


94349 


94398 


49 


8894448 


94497 


94546 


94596 


94645 


94694 


94743 


94792 


94841 


94890 


49 


89 ! 94939 


94987 


95036 


95085 


95133 


95182 


95230 


95279 


95327 


95376 


48 


90 


95424 


95472 


95520 


95563 95616 


95664 


95712 


95700 


95808 


95856 


48 


91 


95984 


95951 


95999 


56047 96094 


96142 


96189 


96236 


96284 


96331 


48 


92 


96378 


96426 


96473 


96520 96567 


96614 


96661 96708 


96754 


96S01 


47 


93 


9684S 


96895 


96941 


96988:97034 


97081 


97127 97174 


97220 


97266 


47 


94 


97312 


97359 


97405 


97451 97497 


97543 


97589 


97635 


97680 


97726 


56 


95 


97772 


9781S 


97863 


97909 197954 


93000 


98045 


98091 


93136 


98181 


56 


9u 


98227 


98272 


98317 


93362i98407 


98452 


98497 


98542 


9S5S7'9S632 


55 


97 


9S677 


98721 


98766 


98811 98855 


9&900 


98945 


98989 


990339907S 


55 


98 


99122 


99166 


99211 


99255:99299 


99343 99387 99431 


9947599519 


54 


99 


99563 


99607 


99651 99694 J9973S 


99782 99825 99869 


9991399956 


54 


N i 





1 


2 1 3 1 4 


5 1 6 1 7 


_8 1 9 _ 


Prop. 


Jfoamplcs. Find the Logarithm of 




Log. 3? ----.. 047712, the answer. 


Log. 54? 1*73239, " 


Log. 867? ...... 2-93802, « 



60 








Logarithms Sure. 








Deg 


0' 


10' i 20' 


30' 


40' 


50' 


60' 







0:00000 


3:46372 3 


26475 


3:94084 


2:06577 


2.-1626S 


2:24185 


89 


1 


2:24185 


2:30879 ] 2 


36677 


2 


41791 


2 


46366 


2:50504 


2:54281. 


88 


2 


2:54281 


2:57756 2 


60973 


2 


63968 


2 


66768 


2:69399 


2:71880 


87 


3 


2:71880 


2:74225 2 


76451 


2 


78567 


2 


80585 


2:82513 


2:84358 


86 


4 


2:84358 


2:86128 


2 


87828 


2 


89464 


2 


91040 


2:92560 


2:94029 


85 


5 


2:94029 


2:95449 


2 


96824 


2 


98157 


2 


99449 


1:00704 


1:01923 


84 


6 


1:01923 


1:03108 


1 


04262 


1 


05385 


1 


06480 


1:07548 


1:08589 


83 


7 


1:08589 


1:09606 


1 


10599 


1 


11569 


] 


12518 


1:13447 


1:14355 


82 


8 


1:14355 


1:15249 


1 


16116 


1 


16970 


1 


17807 


1:18628 


1:19433 


81 


9 


1:19433 


1:20223 


1 


20999 


1 


21760 


1 


22509 


1:23244 


1:23967 


80 


10 


1:23967 


1:24677 


1 


25376 


1 


26063 


1 


26739 


1:27404 


1:28059 


79 


11 


1:28059 


1:28704 


1 


29339 


1 


29965 


1 


30581 


1:31189 


1:31787 


78 


12 


1:31787 


1:32378 


1 


32959 


1 


33533 


1 


34099 


1:34657 


1:35208 


77 


13 


1:35208 


1:35752 


1 


36288 


1 


36818 


1 


37341 


1:37857 


1:38367 


76 


14 


1:38367 


1:38871 


1 


39368 


1 


39860 


1 


40345 


1:40S25 


1:41299 


75 


15 


1:41299 


1:41768 


1 


42231 


1 


42689 


1 


43142 


1:43590 


1:44033 


74 


16 


1:44033 


1:44472 


1 


44905 


1 


45334 


1 


45758 


1:46178 


1:46593 


73 


17 


1:46593 


1:47004 


1 


47411 


1 


47814 


1 


48212 


1:48607 


1:48998 


72 


18 


1:48998 


1:49385 


1 


49768 


1 


50147 


1 


50523 


1:50S95 


1:51264 


71 


19 


1:51264 


1:51629 


1 


51991 


1 


52349 


1 


52704 


1:53056 


1:53405 


70 


20 


1:53405 


1:53750 


1 


54093 


1 


54432 


1 


54768 


1:55102 


1:55432 


69 


21 


1:55432 


1:55760 


1 


56085 


1 


56407 


1 


56726 


1:57043 


1:57357 


68 


22 


1:57357 


1:57668 


1 


57977 


1 


58284 


1 


5S587 


1:58889 


1:59187 


67 


23 


1:59187 


1:594S4 


1 


59778 


1 


60070 


1 


60359 


1:60646 


1:60931 


66 


24 


1:60931 


1:61214 


1 


61494 


1 


61772 


1 


62048 


1:62322 


1:62594 


65 


25 


1:62594 


1:62864 


1 


63132 


1 


63398 


1 


63662 


1:63924 


1:64184 


64 


26 


1:64184 


1:64442 


1 


64698 


1 


64952 


1 


65205 


1:65455 


1:65704 


63 


27 


1:65704 


1:65951 


1 


66197 


1 


66440 


1 


66682 


1:66922 


1:67160 


62 


28 


1:67160 


1:67397 


1 


67632 


1 


67866 


1 


68098 


1:68328 


1:68557 


61 


29 


1:68557 


1:68784 


1 


69009 


1 


69233 


1 


69456 


1:69677 


1:69897 


60 


30 


1:69897 


1:70115 


1 


70331 


I 


70546 


1 


70760 


1:70773 


1:71183 


59 


31 


1:71183 


1:71393 


1 


71601 


1 


71803 


1 


72014 


1:72218 


1:72421 


58 


32 


1:72421 


1:72622 


1 


72822 


1 


73021 


1 


73219 


1:73415 


1:73610 


57 


33 


1:73610 


1:73804 


1 


73997 


1 


74188 


1 


74379 


1:74568 


1:74756 


56 


34 


1:74756 


1:74942 


1 


75128 


1 


75312 


1 


75496 


1:75678 


1:75859 


55 


35 


1:75859 


1:76039 


1 


76217 


1 


76395 


1 


76572 


1:76747 


1:76921 


54 


36 


1:76921 


1:77095 


1 


77267 


1 


77438 


1 


77609 


1:77778 


1:77946 


53 


37 


1:77946 


1:78113 


1 


78279 


1 


78444 


1 


78608 


1:78772 


1:78934 


52 


38 


1:78934 


1:79095 


1 


79255 


1 


79415 


1 


79573 


1:79730 


1:79887 


51 


39 


1:79887 


1:80042 


1 


80197 


1 


80351 


1 


80503 


1:80655 


1:80806 


50 


40 


1:80806 


1:80956 


1 


81106 


1 


81254 


1 


81401 


1:81548 


LS1694 


49 


41 


1:81694 


1:81839 


1 


81983 


1 


82126 


1 


82268 


1:82410 


1:82551 


48 


42 


1:82551 


1:82691 


1 


82830 


1 


82968 


1 


83105 


1.-S3242 


1:83378 


47 


43 


1:83378 


1:83513 


1 


83647 


1 


83781 


1 


83914 


1:S4045 


1:84177 


46 


44 


1:84177 


1:84307 


1 


84437 


1 


84566 


1 


84694 


1:84821 


1:84948 


45 


45 


1:84948 


1:85074 


1 


85199 


1 


85324 


1 


8544S 


1:85571 


1:85693 


44 




60' 


50' 


40' 


30' 


20' 


l0' 


0' 


Dor 


Logarithm Cos 


INE. 


The negative index is noted by two points 


, and must always follow an oppo 


Bite operation to that of the mantissa. 11 


the mantissa is added, subtract 


j the index, and vice versa. 











LOQAKITHMS SlNE. 








fl 


D*g. 


0' 10' 


20' 


30' j 40' 


50' 


60' 

1:86412 




46 


1:85693 


1:85815 


1:85936 


1:860.56 1:86175 


1 


8G294 


43 


47 


1:86412 


1:86530 


1:86617 


1:86763 1:86878 


1 


86993 


1:87107 


42 


48 


1:87107 


1:87220 


1:87333 


1:87445 


1:S7557 


1 


S7667 


1:87778 


41 


49 


1:87778 


1:87887 


1:87996 


1:88104 


1:88212 


1 


88319 


1:88425 


40 


50 


1:88425 


1:88531 


1:88636 


1:88740 


1:88844 


1 


88947 


1:89050 


39 


51 


1:89050 


1:89152 


1:89253 


1:89354 


1:89454 


1 


S9554 


1:89653 


38 


52 


1:89653 1:89751 


1:89849 


1:89946 


1:90043 1 


90139 


1:90234 


37 


53 


1:90234; 1:90329 


1:90424 


1:90517 


1:90611 


1 


90703 


1:90795 


36 


54 


1:90795 1:90887 


1:90978 


1:91068 


1:91158 


1 


91247 


1:91336 


35 


55 


1:91336 1:91424 


1:91512 


1:94599 


1:916S5 


1 


91771 


1:91857 


34 


56 


1:91857 1:91942 


1:92026 


1:92110 


1:92194 


1 


92276 


1:92359 


33 


57 


1:92359 1:92440 


1:92522 


1:92602 


1:92683 


1 


92762 


1:92842, 


32 


58 


1:92842 1:92920 


1:92998 


1:93076 


1:93153 


1 


93230 


1:93306 


31 


69 


1:95306 


1:93382 


1:93457 


1:93532 


1:93606 


1 


93679 


1:93753 


30 


60 


1:93753 


1:93825 


1:93898 


1:93969 


1:94040 


1 


91111 


1:9*1181 


29 


61 


1:94181 


1:94251 


1:94321 


1:94389 


1:94458 


1 


94526 


1:94593 


28 


62 


1:94593 


1:94660 


1:94726 


1:94792 


1:94858 


1 


94923 


1:94988 


27 


63 


1:94988; 1:95052 


1:95115 


1:95179 


1:95241 


1 


95304 


1:95366 


26 


64 


1:95366 ! 1:95427 


1:95488 


1:95548 1:95608 


1 


95668 


1:95727 


25 


65 


1:95727 1:95786 1:95844 


1:9590211:95959 


1 


96016 


1:96073 


24 


66 


1:9607311:96129 1:96184 


1:96239 1:96294 


1 


96348 


1:96402 


23 


67 


1:96402 1 1:96456 1:96509 1:96561 1 1:96613 


1 


96665 


1:96716 


22 


68 


1:96716 1:96767 1:96817 


1:96S67 11:96917 


1 


96966 


1:97015 


21 


69 


1:97015 1:97063 .1:97111 


1:97158 


1:97205 


1 


97252 


1:97298 


20 


70 


1:97298 


1:97344 


1:97389 j 1:97434 


1:97479 


1 


97523 


1:97567 


19 


71 


1:97567 


1:97610 


1:97653 1:97695 


1:97737 


1 


97779 


1:97820 


18 


72 


1:97820 1:97861 


1:97901 1:97942 1:979S1 


1 


9S020 


1:9S059 


17 


73 


1:98059 ! 1:98098 1:98136 1:98173 1 1:98210 


1 


98247 


1:98284 


16 


74 


1:98284 1:9S320 1:98355 1:98391 1:98425 


1 


98460 


1:98494 


15 


75 


1:98494 1:98528 1:9S561 1:98594 ! 1:98626 


1 


98658 


1:98690 


14 


76 


1:98690 1:98721 1:98752 1:98783 1:98813 


1 


98S43 


1:98872 


13 


77 


1:98872 1:98901 1:98930 1:98958 1:98986 


1 


99013 


1:99040 


12 


78 


1:99040 1:99067 1:99093 [ 1:99119 1:99144 


1 


99169 


1:99194 


11 


79 


1:99194 1:99219 1:99243 1:99266.1:99289 


1 


99312 


1:99335 


10 


80 


1:99335 ; 1:99357 i 1:99378 1:99400 1:99421 


1 


99441 


1:99462 


9 


81 


1:99462 ' 1:99481 1:99501 ' 1:99520 1:99539 


1 


99557 


1:99575 


8 


82 


1:99575 1:99592 1:99610 1:99626 1:99643 


I 


99659 


1:99675 


7 


83 


1:99675 1:99690 1:99705 1:99719 1:99734 


1 


99748 


1:99761 


6 


84 


1:99761 1:99774 1:99787 1:99799 1:99811 


1 


99823 


1:99S34 


5 


85 


1:99834 


1:99845 1:99855 1:99865 


1:99875 


1 


99885 


1:99894 


4 


86 


1:99894 


1:99902 1:99911 1:99918 


1:99926 


1 


99933 


1:99940 


3 


87 


1:99940 


1:99946 j 1:99952 1:99958 


1:99964 


1 


99968 


1:99973 


2 


88 


1:99973 


1:99977 1:99981 


1:99985 


1:99988 


1 


99991 


1:99993 


1 


89 


1:99993 


1:99995 1:99997 


1:99998 


1:99999 


1 


99999 


1:99999 







i 60' 


50* 1 40' 


30' 


20' 


10' 


0' 


De*. 


Logarithm Cosine. 


Examples. Find the Logarithms, 


Logrsine 35° 40' ? = 1:76572. the answer. 


Losr:cosine 1S° 20' ? = 1:97737 « 



62 






Logarithms Tangent. 








Dog 


0' 


10' 


20' 


30' 


w 


50' 


60' 


Dug. 





3:00000 


3:46372 


3:76476 


3:94085 


2:06580 


2:16272 


2:24192 


S9 


1 


2:24192 


2:30888 


2:36689 


2:41806 


2:46384 


2:50526 


2.-5430S 


88 


2 


2:54308 


2:57787 


2:61009 


2:64009 


2:66816 


2:69452 


2:71939 


87 


3 


2:71939 


2:74292 


2:76524 


2:78648 


2:80674 


2:82610 


2:84464 


86 


4 


2:84464 


2:86243 


2:87952 


2:89598 


2:91184 


2:92715 


2:94195 


85 


5 


2:94195 


2:95626 


2:97013 


2:98357 


2:99662 


1:00929 


1:02162 


84 


6 


1:02162 


1:03360 


1:04528 


1:05665 


1:06775 


1:07857 


1:08914 


83 


7 


1:08914 


1:09946 


1:10955 


1:11942 


1:12908 


1:13854 


1:14780 


82 


8 


1:14780 


1:15687 


1:16577 


1:17449 


1:18305 


1:19146 


1:19971 


81 


9 


1:19971 


1:20781 


1:21578 


1:22360 


1:23130 


1:23887 


1:24631 


80 


10 


1:24631 


1:25364 


1:26086 


1:26796 


1:27496 


1:28185 


1:28863 


79 


11 


1:28865 


1:29534 


1:30195 


1:30846 


1:31488 


1:32122 


1:32747 


78 


12 


1:32747 


1:33364 


1:33973 


1:34575 


1:35169 


1:35756 


1:36336 


77 


13 


1:36336 


1:36909 


1:37475 


1:38035 


1:38588 


1:39136 


1:39677 


76 


14 


1:39077 


1:40212 


1:40741 


1:41265 


1:41784 


1:42297 


1:42805 


75 


15 


1:42805 


1:43308 


1:43805 


1:44298 


1:44787 


1:45270 


1:45749 


74 


16 


1:45749 


4:46224 


1:46694 


1:47160 


1:47622 


1:48080 


1:48534 


73 


17 


1:48533 


1:48983 


1:49429 


1:49872 


1:50310 


1:50746 


1:5117* 


72 


18 


1:51177 


1:51605 


1:52030 


1:52452 


1:52870 


1:53285 


1:53697 


71 


19 


1:53697 


1:54106 


1:54511 


1:54914 


1:55314 


1:55712 


1:56106 


70 


20 


1:56106 


1:56498 


1:56887 


1:57273 


1:57657 


1:58038 


1:58417 


69 


21 


1:58417 


1:58794 


1:59168 


1:59539 


1:59909 


1:60276 


1:60641 


68 


22 


1:60641 


1:61003 


1:61364 


1:61722 


1:62078 


1:62433 


1:62785 


67 


23 


1:62785 


1:63135 


1:63483 


1:63830 


1:64174 


1:64517 


1:64858 


66 


24 


1:64858 


1:65197 


1:65534 


1-65S70 


1:66204 


1:66536 


1:66867 


65 


25 


1:66867 


1:67196 


1:67523 


1:67849 


1:68174 


1:68496 


1:68818 


64 


26 


1:68818 


1:69138 


1:69456 


1:69773 


1:70089 


1:70403 


1:70716 


63 


27 


1:70716 


1:71028 


1:71338 


1:71647 


1:71955 


1:72262 


1:72567 


62 


28 


1:72567 


1:72871 


1:73174 


1:73476 


1:73777 


1:74076 


1:74375 


61 


29 


1:74374 


1:74672 


1:74968 


1:75264 


1:75558 


1:75851 


1:76143 


60 


30 


1:76143 


1:76435 


1:76725 


1:77014 


1:77303 


1:77590 


1;77877 


59 


31 


1:77877 


1:78163 


1:78447 


1:78731 


1:79015 


1:79293 


1:79578 


58 


32 


1:79578 


1:79859 


1:80139 1:80418 


1:80697 


1:80974. 


1:81251 


57 


33 


1:81251 


1:81527 


1:81803 


1:82078 


1:82352 


1:82625 


1:82898 


56 


34 


1:82898 


1:83170 


1:83442 


1:83713 


1:83983 


1:84253 


1:84522 


55 


35 


1:84522 


1:84791 


1:85059 


1:85326 


1:85593 


1:85860 


1:86126 


54 


36 


1:86126 


1:86391 


1:86656 


1:S6920 


1:87184 


1:87448 


1:87711 


53 


37 


1:87711 


1:87974 


1:88236 


1:88498 


1:88759 


1:89020 


1:89281 


52 


38 


1:S9281 


1:89541 


1:89801 


1:90060 


1:90319 


1:90578 


1:90836 


51 


39 


1:90836 


1.91095 


1:91352 


1:91610 


1:91867 


1:92121 


1:92381 


50 


40 


1:92381 


1:92637 


1:92894 


1:93149 


1:93405 


1:93661 


1:93916 


49 


41 


1:93916 


1:94171 


1:94497 


1:94680 


1:94935 


1:95189 


1:95443 


48 


42 


1:95443 


1:95697 


1:95926 


1:96205 


1:96458 


1:96712 


1:96965 


47 


43 


1:96965 


1:97218 


1:97471 


1:97725 


1:97978 


1:98230 


1:98483 


46 


44 


1:98485 


1:98736 


1:98989 


1:99242 


1:99494 


1:99747 


1:00000 


45 


Peg. 


60' 


50' 


40' 


30' 


20' 


10' 


0' 


I) eg. 


rrv.. 






LOGAR 


THM COTA 


NGENT. 
arm mn 


sf. nl wo va 


rillrkw nrt 


rmnr». 



dte operation to that of the mantissa. If the mantissa is added, subtract 
.he index, and vice versa. 









Logarithms Tangent. 






63 


Deg 


0' 


10' 20' 


30' 1 40' 1 50' j 60 




45 


0-00000 


0-00252 


0-00505 


0-00758 ' 0-01010 1 0-01263 j 0-01516 


44 


46 


0-01516 


0-01769 


0-02022 


0-02275 


0-0252810-02781 


0-03034 


43 


47 


0-03034 


0-03287 


0-03541 


0-03794 


0-04048 


0-04302 


0-04556 


42 


48 


0-04556 


0-04810 


0-05064 


0-05319 


0*05573 


0-05828 


0-06083 


41 


49 


0-06083 


0-06339 


0-06594 


0-06850 


0-07106 


0-07362 


0-07618 


40 


50 


0-07618 


0-07875 


0-08132 


0-08389 


0-08647 


0-08904 


0-09163 


39 


51 


0-09163 


0-09421 


0-09680 


0-09939 


0-10199 


0-10458 


0-10716 


38 


52 


0-10719 


0-10979 


0-11240 


0-11502 


0-11763 


0-12025 


0-12288 


37 


53 


0-12288 


0-12551 


0-12815 


0-13079 


0-13343 


0-13608 


0-13873 


36 


54 


0-13873 


0-14139 


0-14406 


0-14673 


0-14940 


0-15208 


0-15477 


35 


55 


0-15477 


0-15746 


0-16016 


0-16286 


0-16557 


0-16829 


0-17101 


34 


56 


0-17101 


0-17374 


0-17647 


0-17921 


0-18196 


0-18472 


0-18748 


33 


57 


0-18748 


0-19025 


0-19302 


0-19581 


0-19860 


0-20140 


0-20421 


32 


58 


0-20421 


0-20702 


0-20984 


0-21268 


0-21552 0-21S36 


0-22122 


31 


59 


0-22122 


0-22409 


0-22696 


0-22985 


0-23274 0-23564 


0-23856 


30 


60 


0-23856 


0-24148 


0-24441 


0-24735 


0-25031 


0-25327 


0-25624 


29 


61 


0-25624 


0-25923 


0-26222 


0-26523 


0-26825 


0-27128 


0-27432 


28 


62 


0-27432 


0-27737 


0-28044 


0-28352 


0-28661 


0-28971 


0-29283 


27 


63 


0-29283 


0-29596 


0-29910 


0-30226 


0-30543 0-30861 


0-31181 


26 


64 


0-31181 


0-31503 


0-31826 


0-32150 


0-32476 


0-32803 


0-33132 


25 


65 


0-33132 


0-33463 


0-33795 


0-34129 


0-34465 


0-34802 


0-35141 


24 


66 


0-35141 


0-35482 


0-35825 


0-36169 


0-36516 0-36864 


0-37214 


23 


67 


0-37214 


0-37567 


0-37921 


0-38277 


0-38635 0-38996 


0-39359 


22 


68 


0-39359 


0-39723 


0-40090 


0-40460 


0*40831 


0-41205 


0-41582 


21 


69 


0-415S2 


0-41961 


0-42342 


0-42726 


0-43112 


0-43501 


0-43893 


20 


70 


0-43893 


0-44287 


0-44685 


0-45085 


0-45488 


0-45893 


0-46302 


19 


71 


0-46302 


0-46714 


0-47129 


0-47548 


0-47969 


0-48394 


0-48822 


18 


72 


0-48822 


0-49254 


0-49689 


0-50127 


0-50570 


0-51016 


0-51466 


17 


73 


0-51466 


0-51919 


0-52377 


0-52839 


0-53305 


0-53775 


0-54250 


16 


74 


0-54250 


0-54729 


0-55213 


0-55701 


0-56194 


0-56692 


0-57194 


15 


75 


0-57194 


0-57702 


0-58215 


0-58734 


0-59258 


0-S9787 


0-60322 


14 


76 


0-60322 


0-60864 


0-61411 


0-61964 


0-62524 


0-63090 


0-63663 


13 


77 


0-63663 


0-64243 


0-64830 


0-65424 


0-66026 


0-66635 


0-67252 


12 


78 


0-67252 


0-67877 


0-68511 


0-69153 


0-69804 


0-70465 


0-71134 


11 


79 


0-71134 


0-71814 


0-72503 


0-73203 


0-73913 


0-74635 


0-75368 


10 


80 


0-75368 0-76112 


0-76869 


0-77639 


0-78422 


0-79218 


0-80028 


9 


81 


0-80028,0-80853 


0-81694 


0-82550 


0-83422 


0-84312 


0-85219 


8 


82 


0-85219 0-86145 


0-87091 


0-88057 


0-89044 


0-90053 


0-91085 


7 


83 


0-91085 0-92142 


0-93224 


0-94334 


0-95471 


0-96639 


0-97838 


6 


84 


0-97838 


0-99070 


1-00337 


1-01642 


0-02986 


1-04373 


1-05S04 


5 


85 


1-05804 


1-07284 


1-08815 


1-10401 


1-12047 


1-13756 


1-15535 


4 


86 


1-15535 


1-17389 


1-19325 


1-21351 


1-23475 


1-25707 


1-28060 


3 


87 


1-28060 


1-30547 


1-33184 


1-35990 


1-38990 


1-42212 


1-45691 


2 


88 


1-45691 


1-49473 


1-53615 


1-58193 


1-63310 


1-69111 


1-75807 


1 


89 


1-75807 


1-83727- 


1-93419 


2-05914 


1-23523 


2-53627 


0-00000 







60' 


50' 


40' 


30' 


20' 


10' 


0' 


Deg 


LOGARl 


THii Cotangent. 


Example. Find the Logarithms 


> 


Log.tan. 36° 40' ? - 


= 1:87185, the answer. 


Log.tan. 58° 50' ? - 


= 0:21836, " 



64 



Arithmetical Prosre?sion t . 



ARITHMETICAL PROGRESSION. 

Arithmetical Progression is a series of numbers, as 2, 4, 6, 8, 10, 12, 

&c, or 18, 15, 12, 9, 6, 3, in which every successive term is increased or dimin- 
ished by a constant number. 

Letters will denote, 
a = the first term of the series. 
b = any other term whose number from a is n. 
n = number of terms within a and b. 
S = the difference between the terms. 
S = the sum of all the terms. 
In the series, 2, 5, 8, 11, a = 2, b = 11, n = 4, S = 3,and S= 26. 
JgQr'When the series is decreasing, take the first term = b and the last term 
= a. 

The accompanying Table contains all the formulas or questions in Arithmeti- 
cal Progressions, and the nature of the question will tell which formula is to be 
used. 

Formula $ for Arithmetical Progressions. 



a = b—S (n— 1), 
a = &, 






b = a+$ (n— 1), 



& = a, - 

to 2 



2S 



* = 



6 — a 
'n—i' 

(b+g)(]b-a), 

'- ■ 2S—ar-b , 

2(S-an) 
s »(»— 1)/ 

2(5n— S) 
n (n— 1) 



(a-f&)(&+g— a) . 
2* 



a==i 2±\/( 6 +|)-^ " • 

w- 1 a j_. XT* a\ 2 , 2S 

n -2-~±\/(a"-^ + — ' " 
^ 1 j- 6 -l //i M 2 2s 

n== 2" f I±\/(2 + ^--' 



9, 

•10, 

11, 

•12, 



*-»fe±0 f --. 13, 

2 



14* 



£=n[a-f| (n-1)] 15, 
^=n[&_i(n-l)] - -16, 



17, 
18, 
19, 

20, 



Arithmetical Progression. 65 



Example 1, A man was engaged to dig a "well at one dollar ($1) for the first 
foot of the depth of the well, $1*84 for the second, and 84 cents more per every 
successive foot in depth, until he reached the water, which was found at a depth 
of 25 feet. How much money is due to the man ? 

This will be answered by the formula 15, in which a = 1, d = 0*84, and 
n = 25, then the sum, 

S = 25 [ l-f .-^(25 - 1)] = $277 the answer. 

Example 2. A Propeller ship which is to run between Philadelphia' and 
Charleston, cost $116500, of which the company agreed to pay en account 
$14075 at her first trip to Charleston; and per every successive trip, they paid 
$650 less than the former. How many trips must the vessel make until she is 
fully paid ? 

This will be answered by the formula 20, in which o = $14075, d = 650, and 
S = 116500. 



n= o + 



1, 3, 6, 10, 15, 21, . 
1, 4, 10, 20, 35, 56, 



U 1 ^ 5 fnjm 1X^2X116500 = 10 . 6orlltrips . 

2 650 \/ V 650 ^2' 650 

Arithmetical Progressions of a Higher Order. 

Arithmetical Progressions are of the first order, when the difference $ is a 
constant number, but when the difference S progresses itself with a constant 
number, the Progression is of the second order. 

When the difference $ progresses in a second order, the Progression is of the 
third order, &c, &c, and is thus explained : 

1, 2, 3, 4, 5, 6, . . . . n, - • Arith. Prog., first order. 

njn+l) .... 2d. order. 
2 

n(»+l)(n+2>. .... 

2X3 ■ 

1,5,15,35,70,126, . n J^^l±l\ . . . 4th. order. 

Here you will discover that the sum of n terms in one order, is equal to the 
same nth term in the next higher order. Arithmetical Progressions of the first, 
second, and third orders, are applied to 

PILES OF BALLS AND SHELLS. 

Triangular Piling. 
Example 1. A complete triangular pile of balls has n — 12 balls in each side. 
Require how many balls in the base, and how many in the whole pile ? 

In the base, - = - 12 ^ +1) = 78 balls, - - - 2d. order. 
Whole pile, • • SB 1202^ffl2+21 =364 balls, - - 3d. order. 

Square Piling, 
1, 4, 9, 16, 25, 36, - - - - n« 2d. order. 

1,5,14,30,55,91, - - n(n+ PJf +1) , .... 3d. order. 

[See Examples 2 and 3 on page 67.] 
6* 



Geometrical Progression. 



GEOMETRICAL PROGRESSION. 

Geometrical Progression is a series of numbers, as 2 : 4 : 8 : 16 : 32 : &c, 

or 729 : 243 : 81 : 27 : 9 : &c, in which every successive term is multiplied or divided 
by a constant factor. 

Letters will denote, 

a = the first term of the series. 

6 — any other term whose number from a is n. 

n = number of terms within a and b. 

r — ratio, or the factor by which the terms are multiplied or divided. 

tS= Sum of the terms. 

In the series 1 : 3 : 9 : 27 : a — 1, & = 27, n = 4, r = 3, S = 40. 

The accompanying Table contains all the formulas or questions in Geometrical 
Progressions. The nature of the question will tell which formula is to be 
used. 

Formulas for Geometrical Progressions. 



« = -, 

rn—y 

a^S—r(S—b),. 
r— 1 . 



a = # 



r*—l> 



5— a 



• - 7, 

-^6' " * * 8 ' 
ar»4-,$r— rS — a = 0, - 9, 



b = ar*-\ • 4, 

h = S - S —*, . . 6 , 
r 



£= 



5r — a 



r — l> 



S= 



a(r» — 1) 



-11, 






_ , loa.b — Toq.a 
log.r ■ 



?•-*+» 



6ff.fr — 7off.a 



Zo^.ra+^r —1 )] — ftff.a • - -- 
fty.r. » 15 ' 

• - - 16, 



= 2 . 1og.b — log.{br -r- ff(r— 1) ] 



£= 






W, 



Geometrical Progression. 67 



Examplel. Required the 10th term in the Geometrical Progression 4 : 12 : 36..., ? 

Given a == 4, tj = 10, and.r = 3. "We have, 

Formula 4. 6 = ar»-i = 4X3 9 = 78732, the tenth term. 

Example 2. Required the sum of the 10 terms in the preceding example ? 

Formula 11, S = 2£=i)__ 4(3io— 1) = ug0 |fafl gum> 

' r — 1 2 

Example 3. Insert 6 proportional terms "between 3 and 384 1 
Given a = 3, & = 384, and n = 6+2 = 8. 

Formula 7, r 

then 3 : 6 : 12 : 24 : 48 : 96 : 192 : 384, the answer. 

Example 4. A man had 16 twenty dollar gold pieces, which he agreed to ex- 
change for copper in such a way, that he gets one cent on the first $20, two on 
the second, four on the third, and eight on the fourth, &c, &c; until the sixteen 
$20 pieces were covered. How many cents will come on the sixteenth gold 
piece, and what will he the whole amount of copper on the gold? 

In the progression 1:2:4:8: &c, we have, 
Given n = 16, r =2, and a = 1, then, 

» i 2 15 48 16* 2563 
Formula 4. b = 1X2 16 " 1 -=— = -=--==— = 32768 cents, on the 

4 2 2-2! 

sixteenth piece. 
The total sum of cents will he found hy the 

Formula 10, £- 3 27 ^X2-- l = 65535 cents = $655-35. 



Piling of BaUs and Shells. — [From page 64.] 

Example 2. How many halls are contained in a complete square pile, n = 10 
rows? 

10(10+1)(2X10+1) = 10X11X21 =385 Ms< 
2X3 6 

Rectangular Filing. 

Let m he the numher of halls on the top of the complete pile, and n = num- 
ber of rows in the same, then the numher of halls in the whole pile will 
be, 

n(n-\-l)(2n+Zm—2) 

2 X 3 " " " Sd ' order - 

The numher of halls in the longest bottom side will be = m+n — 1. 

Example 3. The rectangular pile having 15 rows and 23 balls on the the top, 
how many in the whole pile ? 

1 5(15+l)(2Xl5-f3X23-2) 15 X 16X67 ORfin ... 



63 Compound Interest.— Annuities. 




COMPOUND INTEREST. 




Compound Interest 3 

each year, and the sum is the 
Amount, 


s when the Interest is added to the 
Capital for the following yeai. 
a = c(l-hp)», - - - 


Capital for 


Capital, 


a 


m 1 


c q.+p>' 


' *9 


Per centage, 


,-#r-i, . . . . 

c 


• 3, 


Number of years, 


log.a — log.c 9 
n ~ log.(l+p) 


- 4, 



j&§=*In these formulas p must be expressed in a fraction of 100. 
Example 1. A capital c = 8650 standing with Compound Interest at p = 5 
per cent, what will it amount to in n = 9 years. 
Amount a = 8650 (1-05)9 — 13419 dollars. 

Example 2. A man commenced business with c = 300 dollars, after n = 5 
years he had a = 6875 ^dollars At what rate did his money increase, and 
how soon will he have a fortune of 50000 dollars ? 

The first question, or the per centage, will be answered by the formula 3. 

* = v ^f — 1 = <£/ 22-9166 — 1 = 0-87, or 87 per cent. 

The time from the commencement of business until the fortune is completed, 
will be answered from the formula 4. 

%.50000 — Zoor.300 4-69897—2-47712 - „ 
»= fty.1-87 " 0-2720048. " Mfl9 W 

or 8 years and 2 months. 



ANNUITIES. 

Annuity is a certain sum of money to be paid at regular intervals. 
A yearly payment or annuity b, is standing for n years, to find the whole 
amount a at p per cent. Interest. 

Amount, a = bn I l+4j (**+*) J Simple Int., 1, 

Amount, a = — I (l+j?) n — lj Comp. Int., - • - - 2, 

A yearly payment or annuity 5, is to be paid for n years, to find the present 
worth, or the amount a, which would pay it in full, at the beginning of the 
time n, deducting p per cent. Interest. 

Amount, a = &n[l — ^(-^JjSimp. Int., • - - 3, 
Amount, a — — X 1 — — J Comp. Int., ^ • 4, 



Annuities.— Paper.— Selection of "Water Colours. 



69 



A debt D, standing for Interest, is diminished yearly by a sum b ; to find the 
debt d after n years, and the time n when it is fully paid? 
The debt d after n years will be, 

,.^-^w^^ . . . 6> 

The time n until fully paid will be, 

k>g.o — log.(b~-Dp) . 

U = &V.0L+P) 6 ' 

If 5 = Dp then n = 00, or the debt D will never be paid. If 6<.Pp, the debt 
D will be increased. 

To find the yearly annuity 6, which will pay a debt D in n years, at p per 
cent. Compound Interest ? 





(i-kP)»-l 


• • 4» 




PAPE*. 






1 Ream = 20 quires = 480 sheets. 






1 quire = 24 sheets. 






Drawing Paper. 




Cap, 


- 13X16 inches. 


Columbier, 


34X23 inches. 


Demy, 


20X15 " 


Atlas, - 


33X26 " 


Medium, 


. 22X17 « 


Theorem, - 


34X28 « 


Royal, 


24X19 " 


Double Elephant, 


40X26 " 


Super Royal, - 


• 27X19 " 


Antiquarian, - 


52X31 " 


Imperial, - 


80X21 " 


Emperor, - 


40X60 " 


Elephant, 


- 28X22 " 


Uncle Sam, - 


48X120 " 



Glazed or Crystal, 
Yellow or Blue Wove. 



Continuous Colossal Drawing Paper, No. A, and No. B, 56 inches wide, and of 
any required length. No. A, of this paper is excellent for mechanical drawings. 
Price from 40 to 50 cents per yard. 

Tracing Paper. 
Double Crown, 30 by 20 inches.") 

Double Double Crown, 40 " 30 ." > 

Double Double Double Crown, 60 " 40 " J 

Finest French Vegetable Tracing Paper. 
Grand Raisin (or Royal) 24 in. by 18. Grand Aigle 40 in. by 27. 

, Mounted Tracing Paper. 

This paper is mounted on cloth, and is still transparent ; it will take ink and 
water colours. It is 38 inches wide, and of any required length. 
YeUum Writing Cloth, 
Adapted for every description of tracing ; it is transparent,durable, and strong. 
It is 18 to 38 inches wide, and of any required length. 



SELECTION OF WATER COLOURS. 



Blue. Real Ultramarine. 
" French Blue. 
" Indigo. 
" Cobalt Blue. 
Green. Olive Green. 
TeUovi. Cadmium. 
" Gamboge. 
" Ochre. 
Bed. Carmine. 
" Crimson Lake. 



Bed. Rose Madder. 

" Light Red. 

Brown. "Vandyke. 

" Brown Madder. 

Black. India Ink. 

" Blue Black. 

" Ivory Black. 

" Lamp Black. 
White. Chinese White. 



70 



United States' Standard Measures and Weights. 



UNITED STATES' STANDARD MEASURES AND WEIGHTS. 

MEASURE OF LENGTH. 
The Standard Measure of Length is a brass rod = 1 yard at the temperature of 
32° Fahrenheit. The length of a pendulum vibrating seconds in vacuo, at 
Philadelphia is 1*08614 yards, at + 32° Fahrenheit. 

The Surveying Chain is = 22 yards = 66 feet. It consists of 100 links, 
and each link = 7*92 inches. 

ROPES AND CABLES. 
1 Cable length = 120 fathoms = 720 feet. 
1 fathom = 6 feet. 

GEOGRAPHICAL AND NAUTICAL MEASURES. 
1 Degree of the great circle of the Earth round the Equator = 69-032 statute 
miles as 60 Nautical miles. 

1 Statute mile = 5280 feet = 0-86875 Nautical miles. 
1 Nautical mile = 6037*424 = 1-150 Statute miles. 

LOG LINE. 

The IiOg Line should be about 150 fathoms long, and 10 fathoms from 
the Log to the first knot on the line. If half a minute glass is used, it will be 
51 feet between each succeeding knot. For 28 seconds glass it will be 47*6 feet 
= 7*93 fathoms per knot. This is the length of knot by calculation, but prac- 
tically it is shortened to 7 "5 fathoms per knot for 2S seconds glass. 

MEASURE OF CAPACITY. 

Gallon* The standard Gallon measures 231 cubic inches, and contains 
8-33SS822 pounds Avoirdupois = 58372-1757 grains Troy, of distilled water, at its 
maximum density 39*83° Fahrenheit, and 30 inches barometer height. 

Bushel* The standard Bushel measures 2150*42 cubic inches = 77*627413 
pounds Avoirdupois of distilled water at 39*83° Fahrenheit, barometer 30 inches. 
Its dimensions are 18£ inches inside diameter, 19£ inches outside, and 8 inches 
deep ; and when heaped, the cone must not be less than 6 inches high, equal 
2747*70 cubic inches for a true cone. 

Pound. The standard Pound Avoirdupois is the weight of 27*7015 cubic 
inches of distilled water, at 39*83° Fahrenheit, barometer SO inches, and 
weighed in the air. 

MEASURE OF LENGTH. 



Miles. 

1 

0125 

00125 

0003125 

000056818 

000018939 

O 000015783! 



Furlongs. 



Chains. 



025 



10 

1 



01 

0025 

00045454 0*045454 

000151515 01515151 

;0-000120262 0*001262626, 



Rods. 

320 

40 

4 

1 

0-181818 

00606060 



Yards. 

1760 
220 
22 
5.5 

1 
0*33333 



0*00505050 ,0*0277777 



Feet. 

5280 

660 

G6 

16*5 

3 

1 

0083333 



Inches. 

G3360 
7920 

792 

198 

36 

12 

1 



MEASURE OF SURFACE. 



Sq. Miles. Acres S.Chams. Sq. Rods. 'Sq. Yards. Sq. Feet. Sq. Inches. 



1 

0001562 

00001562 

0000009764 

0000000323 

0-0000000358 



640 
1 

0*1 

0-00625 

00002066 

0000002296 



6400 
10 
1 
00625 
0002066 



102400 

160 

16 

1 

0-0330 



0-00002296,000367 



3097600 

4840 

484 

30*25 

1 

0*1111111 



27878400 
43560 
4356 
272-25 

9 

1 



0-00000000025,0000000143 l 000000143 l 0*00002552.20-0007716 0006944 



4014489600 

696960 



39204 
1296 
144 

1 





United States' Standard Measures and Weights. 


fl 


MEASURE OP CAPACITY. 


Cub. Yard. 


Barrels. 


Bushels. 


Cub. Feet. 


Pecks. 


Gallons. 


Cub. Inch 


1 
0-1782 
003961 
0-037037 
0-009902 
0-004951 
000002143 


5-6103 
1 
02222 
0-2078 
005555 
002777 
0001202 


25-2467 
45 
1 
0-804 
0-25 
0125 
0-000465 


27 
4-8125 
1-2438 
1 
0-26738 
0-13369 
0-0005787 


100-987 201-974 

18 36 

4 8 

3-73809 7-47619 

1 2 

05 1 

00021645 L 0004329 


46656 
8316 
2150-42 
1728 
462 
231 
1 


MEASURE OF LIQUIDS. 


Gallon. 


Quarts. 


Pints. 


Gills. 


Cub. inch. 


1 

0-25 
0125 
003125 
0-004329 


4 

1 
0-5 
0-125 
0017315 


8 
2 
1 

0-25 

003463 


32 

8 

4 

1 

0-13858 


231 
57-75 

28-875 

7-2175 

1 




MEASURES OF WEIGHTS. 




AVOIRDUPOIS. 


Ton. 


Cwt. 


Pounds. 


Ounces. 


Drams. 


1 
005 

0-00044642 
00002790 
00000174 


20 
1 
00089285 
0*000558 

0-0000348 


2240 

112 

1 

0-0625 
0-0016 


35840 

1792 

16 

1 

0-0625 


573440 

28672 

256 

16 

1 


TROY. 


Pounds. 


Ounces. 


Dwfc 


Grains. 


Pound Avoir. 


1 

0-083333 
004166 
0-0001736 
1-215275 


12 

1 
0-05000 
0-00208333 
14-58333 


240 

20 
1 

0-0416666 
219-6666 


5760 

480 

24 

1 

7000 


0-822861 

0068571 
00034285 
000020571 

1 


APOTHECARIES'. 


Pounds. 


Ounces. 


Drams. 


Scruples. 


Grains. 


0-08333 
0-01041666 
0-0034722 
000017361 


12 
1 

0-125 

00416066 

0-020833 


96 

8 

1 

0-3333 

0-16666 


288 

24 

3 

1 

005 


5760 
480 
60 
20 
1 



United States' Standard Measures and Weights. 



Carat. 

1 
0-25 

0015625 
0-3125 



DIAMOND. 
Grain. Parts. 



4 
1 

00625 
1-25 



64 
16 
1 
20 



Grains. Troy. 

32 

0-8 
005 

1 



GOLD COINS. U. S. STANDARD WEIGHT. 



Name of the Coins 

Double Eagle . . . 

Eaa:le 

Half Eagle . . . . 
Three Dollar piece • . 
Quarter Eagle . • . 
Dollar piece .... 
Value per Grain . . 
Value per Ounce . . . 



WEIGHT TROY. 



Dollars. 


Grains. 


Ounces. 


$ 20 


516 


1075 


$ 10 


258 


05375 


$5 


129 


026875 


$ 3 


77-4 


016125 


$ 250 


645 


0-134375 


S 1 


258 


005375 


$ 00387596 


1 


000208333 


$ 18C046 


480 


1 



SILVER COINS. U. S. STANDARD WEIGHT. 



Name of the Coins. 

One Dollar . . ... 

Half Dollar or fire Dimes . 
Quarter Dollar or 2£ Dimes 

One Dime 

HalfDime 

Three Cents piece . . . 
Value per Grain . . . • 
Value per Ounce . . . 



Copper Cent 
Half Cent . . 
Value per Grain , 
Value per Ounce 



WEIGHT TROT. 



Cents. 


Grains. 


Ounces. 


100 


384 


0-8 


50 


192 


0-4 


25 


96 


0-2 


10 


384 


008 


5 


19-2 


004 


3 


11-52 


0024 


0-26041668 


1 


000208333 


125 


480 


1 


1 


168 


0-35 


05 


84 


0175 


000595238 


1 


000208333 


2-8571424 


480 


1 



The Standard fineness of Gold and Silver Coins is one weight of aUoy to 

nine weights of pure metal. The alloy for Gold Coin is Silver and 

Copper, and Copper for Silver Coin. 

Relative value of Foreign Gold and Silver Coins, fixed oy the law of the 
United States. 

] Pound Sterling of Great Britain $ 4*84 

1 Shilling 0242 

1 Pound Sterling of Nova Scotia, New Brunswick, Newfoundland and 

Canada 400 

1 Dollar of Mexico, Peru, and Central America 100 

1 Pagoda of India 1*84 

1 Real Vellon of Spain 0-05 

1 Real Plate of Spain • 0-10 

1 Rupee Company 0-44? 

1 Rupee of British India 044£ 

J Franc of France and Belgium ®' l %%c 

1 Specie Dollar of Sweden and Norway TOO 

1 Ducat of Sweden 2*15 

1 Specie Dollar of Denmark 105 

1 Florin of Netherland 0.40 

1 Florin of Southern States of Germany • . . 0.40 

1 Guilder of Netherland 0.40 

1 Livre Tournoise of France 0.18$ 





Foreign Weights and Measures. 


73 


1 LCvre of the Lonibardy Venitian Kingdom • . 


$0.16 


1 Irivre of Tuscany 


0.16 


1 Livre of Sardinia .... ...., 


. 018?_« 


1 Milrea of Portugal 


1.12 


1 Milrea of Azores • 


. 0.83|- 


1 Marc Banco of Hamburg 


0-35 
. 0.69 

0.78| 
. 0.75 


1 Rix Dollar or Thaler of Prussia and the Northern States of Germany, 
1 Rix Dollar of Brfimpn ______ 


1 Rouble Sih 


v.r of Russia .-._ 


1 Florin of Austria 


0.484 


1 Ducat of Naples 


. 0.80 


1 Ounce of Si 


cilv .__ 


2.40 


1 Tad, of China • . . . . 










Places. ' 


Measures. 'Inches. 


Places. 


Measures. 


Incites. 


Amsterdam 


Foot 


11-14 


Malta . . 


Foot 


11-17 


Antwerp 


.< 


11-24 


Moscow 




a 


13-17 


Bavaria . . 


tt 


11-42 


Naples . 




Palmo 


10-38 


Berlin . . 


tt 


12-19 


Prussia 




Foot 


12-36 


Bremen . . 


a 


11-38 


Persia . 




Arish 


38-27 


Brussels 


" 


11-45 


Rhineland 




Foot 


12-35 


China 


" Mathematic 


13-12 


Riga . 




<* 


10-79 


.. 


" Builder's 


12-71 


Rome . 




a 


11-60 


a 


" Tradesman's 


13-32 


Russia . 




it 


13-75 


" . . 


" Surveyor's 


12-58 


Sardinia 




Palmo 


978 


Copenhagen 


tt 


12-35 


Sicily . 




" 


9-53 


Dresden 


a 


11-14 


Spain . 




Foot 


11-03 


England 


u 


12-00 


" . . 




Toesas 


66-72 


Florence . 


Braccio 


21*60 


u 




Palmo 


8-34 


France . . 


Pied de Roi 


12-79 


Strasburgh 




Foot 


11-39 


u # # 


Metre 


39-3S1 


Sweden 




" 


11-69 


Genera . . 


Foot 


19-20 


Turin . 




it 


12-72 


Genoa . . 


Palmo 


9-72 


Venice . 




a 


13-40 


Hamburgh . 


Foot 


11-29 


Vienna 




" 


12-45 


Hanover 


tt 


11-45 


Zurich . 




tt 


11-81 


Leipsic . . 


u 


11-11 


Utrecht 




" 


10-74 


Lisbon . . 


tt 


12-96 


Warsaw 




tt 


14-03 


'* . . . 


Palmo 


8-64 








ENGLISH AND FRENCH MEASURES OF LENGTH. " 




British. Yard is referred to a natural standard, -which is the length of a 


pendu- 


lum vibrating seconds in vacuo in London, at the level of t 


he sea; 


measured on a brass rod, at the temperature of 62° Fahr 


enheit, 


= 39*1393 Imperial inches. 




French. Old System.— -1 Line = 12 points . = 0-08884 U. S. iu 


ches. 


llnch -___- 12 lines . = 1-06604 " 




1 Foot = 12 inches . = 12-7925 " 




lToise = 6 feet . =76*765 " 




1 League = 2280 toises • (common.) 
1 League = 2000 toises . (p° s t.) 






1 Fathom — 5 feet. 




New System.— 1 Millimetre . . = '03939 TJ. S. in 


ches. 


1 Centimetre . . = -39380 '< 




1 Decimetre = 3-93809 « 




1 Metre . . . . = 39-38091 " 




1 Decametre. . . = 393-80917 rt 




lHecatometre . • =3938-09171 " 





74 




Foreign Weights and Measur. 


S3. 




FOREIGN ROAD MEASURES COMPARED WITH AMERICAN. 


Places. 


Measures 


Yards. 


Places. 


Measures. 


Yards. 


Arabia . 


» 


Mile 


2148 


Hungary . 


Mile 


9113 


Bohemia 




a 


10137 


Ireland 


a 


3038 


China . 




Li 


629 


Netherlands 


(l 


1093 


Denmark 




Mile 


8244 


Persia . . 


Parasang 


6086 


England 


. 


" Statute 


1760 


Poland 


Mile, long 


8101 


" 




" Geographical 


2025 


Portugal . 


League 


6760 


Flanders 


. 


cc 


6S09 


Prussia 


Mile 


8468 


France . 


• 


League, marine 


6075 


Rome . . 


a 


2025 


cc 




" common 


4861 


Russia . . 


Terst 


1167 


a 


. 


" post 


4264 


Scotland . 


Mile 


1984 


Germany 


. 


Mile, long 


10126 


Spain . . 


League, common 


7416 


Hamburgh . 


u 


8244 


Sweden 


Mile 


11700 


Hanover 


. 


" 


11559 


Switzerland 


u 


9153 


Holland 




" 


6395 


Turkey 


Berri 


1826 




MEASURES OF SURFACE. 


French. 


Old System.— 1 Square Inch . . = 1-1364 U. S. inches. 




1 Arpent (Paris) . . = 900 square toises. 




1 Arpent (woodland) = 100 square royal perches. 




New System.— 1 Arc =100 square metres. 




1 Decare . . . . ^ 10 ares. 




1 Hecatare . . . . =• 100 ares. 




1 Square Metre . . — 1550*86 square inches, or 




10-7698 square feet. 




1 Arn - - - - - ^ 1fY7fi-G« a 


FOREIGN MEASURES OF SURFACE COMPARED WITH AMERICAN. 


Places 




Measures. 


Sq. Yds. 


Places. 


Measures. Sq. Yds. 


Amsterdam 


Morgen 


9722 


Portugal . 


Geira 


6970 


Berlin . 




" great 


6786 


Prussia 


Morgen 


3053 


a 




" small 


3054 


Rome . . 


Pezza 


3158 


Canary Isles 


Fanegada 


2422 


Russia . . 


Dessetina 


L3066-6 


England 




Acre 


4840 


Scotland . 


Acre 


6150 


Geneva 




Arpent 


6179 


Spain . f . 


Fanegada 


5500 


Hamburgh . 


Morgen 


11545 


Sweden 


Tunneland 


5900 


Hanover 




u 


3100 


Switzerland 


Faux 


7855 


Ireland . 


. 


Acre 


7840 


Vienna 


Joch 


6889 


Naples . 


. Moggia 


3998 


Zurich 


Common acre 


3875-0 




FOREIGN MEASURES OF CAPACITY. 


British. 


The Imperial gallon measures 277-74 cubic inches, containing 10 lbs. 




Avoirdupois of distilled water, weighed in air, at the temperature 




of 62° degrees, the barometer at 30 inches. 




Far Grain. 8 bushels = 1 quarter. 




1 quarter = 10-2694 cubic feet. 




Coal, or heaped measure. 8 bushels = 1 sack. 




12 sacks = 1 chaldron. 




Imperial bushel = 2218-192 cubic inches. 




^Heaped bushel, 19£ ins. diam., cone 6 ins. high = 2812-4872 cubic ins. 




1 chaldron = 58*658 cubic feet, and weighs 3136 pounds. 




1 chaldron (Newcastle) = 5936 pounds. 


French. 


JS T eiv System. — 1 Litre — 1 cub. decimetre, or 61-074 U. S. cubic inches. 




Old System.— 1 Boisseau = 13 litres = 793-964 cub. ins., or 3-43 galls. 




1 Pinte = 0-931 litres, or 56-817 cubic inches. 


Spanish. 


1 Wine Arroba = 4-2455 gallons. 




1 Fanega (common measure) = 1-593 bushels. 




* When heaped in the form of a true cone. 





Foreign Weights and Measures. 




75 


FOREIGN LIQUID MEASURES COMPARED WITH AMERICAN, 


Places. 


Measures. 


Cub. In. 


Places. i Measures. 


Cub. In- 


Amsterdam . 


Anker 


2331 


Naples . . • 


Wiue Barille 


2544 


" 


Stoop 


146 


u . 






Oil Stajo 


1133 


Antwerp 


a 


194 


Oporto . 






Almude 


1555 


Bordeaux . . 


Barrique 


14033 


Rome . 






Wine Barille 


2560 


Bremen . . 


Stubgens 


194-5 


u 






Oil " 


2240 


Canaries . . 


Arrobas 


949 


It 






Boccali 


80 


Constantinople 


Almud 


319 


Russia . 






Weddras 


752 


Copenhagen . 


Anker 


2355 


u 






Kunkas 


94 


Florence . . 


Oil Barille 


1946 


Scotland 






Pint 


103-5 


" . . 


Wine " 


2427 


Sicily 






Oil Caffiri 


662 


France . . 


Litre 


61-07 


Spain . 






Azumbres 


22-5 


Geneva . . 


Setier 


2760 


a 






Quartillos 


30-5 


Genoa . . 


Wine Barille 


4530 


Sweden 






Eimer 


4794 


" * 


Pinte 


90-5 


a 






Kanna 


159-57 


Hamburgh 


Stubgen 


221 


Trieste 






Orne 


4007 


Hanorer . . 


" 


231 


Tripoli 






Mattari 


1376 


Hungary . 


Eimer 


4474 


Tunis . 




1 ;on <• 


1157 


Leghorn . . 


Oil Barille 


1942 


Venice 




Secchio 


628 


Lisbon . 


Almude 


1040 


Vienna 




. [Eimer 


3452 


Malta . . . iCamri 


1270 


" 




. Maas 


86-33 


FOREIGN DRY MEAS 


URES C( 


)MPARED WITH AMERICAN. 


Places. 


Measures. 


Cub. In. 


Places. "7 Measures. Cub. In. 


Alexandria . 


Rebele 


9587 


Malta . . . 


Salme 


16930 


a u 


Kislos 


10418 


Marseilles 




Charge 


9411 


Algiers . . 


Tarrie 


1219 


Milan . . . 




Moggi 


8444 


Amsterdam 


Mudde 


6596 


Naples 






Tomoli 


3122 


" 


Sack 


4947 


Oporto 






Alquiere 


1051 


Antwerp 


Viertel 


4705 


Persia 






Artaba 


4013 


Azores . 


Alquiere 


731 


Poland 






Zorzec 


3120 


Berlin . . 


Seheffel 


3180 


Riga . . 






Loop 


3078 


Bremen . 


" 


4339 


Rome . . 






Rubbio 


16904 


Candia . . 


Charge 


9288 


a 






Quarti 


4226 


Constantinople 


kislos 


2023 


Rotterdam 






Saeh 


6361 


Copenhagen 


Toende 


8489 


Russia 






Chetwert 


12448 


Corsica . . 


Stajo 


6014 


Sardinia . 






Starelli 


2988 


Florence 


Stari 


1449 


Scotland . 






Firlot 


2197 


Geneva . . 


loupes 


4739 


Sicily . . 






Salme gros 


21014 


Genoa . . 


Mina 


7382 


u 






" generale 16S86 


Greece . . 


Medimni 


2390 


Smyrna . 






Kislos 


2141 


Hamburgh . 


Seheffel 


6426 


Spain . . 






3atrize 


41269 


Hanover 


Vlalter 


6868 


Sweden . 






Tunna 


8940 


Leghorn 


Stajo 


1501 


Trieste 






Stari 


4521 


" . Sacco 
Lisbon . . jAlquiere 


4503 


Tripoli . 






Caffiri 


19780 


817 


Tunis . . 






K 


21S55 


" . * ( Fanega 


8268 


Venice 






Stajo 


4945 


Madeira . . ! Alquiere 


684 


Vienna . 






Metzen 


3753 


Malaga . . .Fanaga 


3783 








FRENCH 


MEASUI 


,ES OF SOLIDITY. 


French. 1 Cubic Foot 


. 


= 2093-470 U. S. inches. 


Decistre 




= 3-5375 cubic feet. 


Stere (a cubic metre) 


. 


— 25-375 " 


Decastere . 




= 353-75 " 


1 Stere 




= 61074-664 cubic inches. 


For the Square and Cubic Mec 


isures of 


other countries, take the length of the 


measure in Table, page 72, and 


square o 


r cube it as required. 



76 


Foreign Weights asp Measubhs. 




ENGLISH AND FRENCH MEASURES OP WEIGHT. 


British. 1 troy Grain = -003961 cubic inches of distilled water. 


1 troy Pound = 22*815689 cubic inches of water. 


Fez>th. Old System. — 1 Grain 


. = 0-8188 grains tror. 


1 Gros 


. = 5S-954S " 


1 Ounce 


. = 1-0780 oz. avoirdupois. 


1 Livre 


. = 1-0780 lbs. " 


Hew /System. — Milligramme 


. = -01543 troy grains. 


Centigramme 


t • = -15433 " 


Decigramme 


. = 1-54331 " 


Gramme 


. = 15-43315 « 


Decagramme 


. = 154-33159 " 


Hecatogramme 


. = 1543-3159 " 


1 Millier = 1000 Kilogrammes = 1 ton sea weight. 


1 Kilogramme 


. = 2-204737 lbs. avoirdupois. 


1 Pound avoirdupois . = 0*4535685 Kilogramme 


1 Pound troy 


. =0-3732223 « 


Note. — In the new French system, the 


values of the base of each measure, viz., 


Metre, Litre, Store, Are, and Gramme, are decreased or increased by Ute foUowing 


words prefixed to tfiem. Thus, 




Milli expresses the 1000th part. 


Hecato expresses 100 times the value. 


Centi " 100th " 


Chilio " 1000 " 


Deci " 10th " 


Myrio " 1000 " 


Deca " 10 times the value. 




FOREIGN WEIGHTS COM! 


5 ARED WITH AMERICAN. 






Number 






JS umber 






equal to 






equal to 


Places. 


Weights. 


100 avoir- 
dupois 
pounds. 


Places. 


Weights. 


100 avoir- 
dupois 
pounds. 


Aleppo 
■• • • 


Rottoli 


20-46 


Hanover 


Pound 


93-20 


Oke 


S5-S0 


Japan 




Catty 


76-02 


Alexandria . 


Rottoli 


107' 


Leghorn 




Pound 


133-56 


Algiers . . 
Amsterdam 


cc 

Pound 


84- 
91-8 


Leipsic . 
Lyons 




" (common) 
" (silk) 


97-14 
98-81 


Antwerp 
Barcelona • 


u 


96-75 


Madeira . 




ss 


143-20 


ss 


112-6 


Mocha . 




Maund 


33-33 


Batavia . • 


Catty 


76-78 


More a . 




Pound 


90-79 


Bengal • # 


Seer 


53-57 


Naples . 




Rottoli 


50-91 


Berlin . • 


Pound 


96.8 


Rome 




Pound 


133-69 


Bologna . . 


cc 


125-3 


Rotterdam 




u 


91-80 


Bremen . # 


ss 


90-93 


Russia . 




ss 


110-86 


Brunswick . 


K 


97*14 


Sicily 




ss 


142-55 


Cairo . • 


Rottoli 


105. 


Smyrna . 




Oke 


36-51 


Candia • . 


a 


S5-9 


Sumatra . 




Catty 


35-56 


China . . 


Catty 


75-45 


Sweden . 




Pound 


106-67 


Constantinople 


Oke 


35"55 


EC 




" 


120-68 


Copenhagen 
Corsica . . 


Pound 
u 


90-80 
131-72 


Tangiers 
Tripoli . 




" (miner's) 
Rottoli | 


94-27 

89-28 


Cyprus . . 
Damascus . 


Rottoli 


19-07 


Tunis . 




'• 


90-09 




25-23 


Venice . 




Pound (he aw) 


94-74 


Florence 


Pound 


133-56 


ss u 




" (light) 


150. 


Geneva . 


'• (heavy) 


82-35 


Tienna . 




CI 


81- 


Genoa . . 


ss V CC 


92-86 


Warsaw . 




Si 


112-25 


Hamburgh . 


H is 


1 93-63 









Provisions. 



go »*j 



Beef. 



Pork. 



Flour. 



Kaisins or 
dried fruit. 



Pickles. 



Rice. 



CQ^^^^^tf^tf^ 



to tO to to to to tO 



Ha W-« W- 1 iW-« 



*-S- iN-« rf4-" 



Sugar. 


Tea. 




Coffee. 




Cocoa. 





#*- ^i- tf^ 



Biscuits. 



Butter. 



Cheese. 



Molasses'. 
Tinegar. 

Spirits. 



2 I 



'Calculus of Differentials. 



By the Differential Calculus we ascertain the simultaneous progress 
of variable quantities depending on one another. The variable quanti- 
ties are designated by the last letters u, v, x, y, z, and the constant 
quantities by the first, a, b, c, e, /, of the alphabet. The letter d is placed 
before variables to denote the instantaneous progress of that quantity, 
as dx, and called the differential of a?, d' reads differential. Let the side 
of a square be denoted by x and the area by % \ when x increases uni- 
formly, z will increase more rapidly. When x = l,z = l, but when x = 2, 
z = 4. When we know the instantaneous increase of #, what will be 
that of zl If we add, say only a point to the side x, there will be added 
two lines or 2 a? to the square. We know that z = x", the d' or increment 
of the square will be dz = 2xdx, of which dx is the point added to x and 
2x is the two lines added to the square, called the differential coefficient. 
Let v denote the volume of a cube, and x its side, we have v = x 3 and 
dv — 3x"dx, which shows that if a point dx is added to x there will be 3a; 3 
or three squares added to the cube. 

The d- of any power of a variable is equal to the power diminished by 
1, multiplied by the primitive exponent and the product by the d' of the 
variable. The d- of a constant is = o. When the constant is a factor to 
the variable it appear unchanged in the d' coefficient, but when a term 
it disappears. 

I. The d' of length u of any line defined by a formula of rectangular 
co-ordinates x and ?/, is die = s /dx--\-dy' 2 . II. The d' of area z of any 
plan figure bounded by a curved line and rectangular co-ordinates in 
dz — ydXt y = ordinate, x = abscissa. III. The d' of solidity v of any 
figure bounded by a plan rotating round its abscissa a?, is dv = n]/ i dx, 
2/ = ordinate of the outer line of the plan. IV. The d' of surface z of 
any solid bounded by a plan rotating round its abscissa x, is dz = 2nydu, 
in which u = length of the outer edge of the plan. 

Successive d-s is when the first d' coeff. is considered a function of a 
new function. 

u = ax\ 1st. d' coeff. ^ = 4^^, 2nd. d- coeff. d{^j =|^ = 12ft ^, 

™ ?fd-u\ d 3 u 
3d. d' coeff. d / — 1 = — = 2 4 a x, etc., etc. 

d-y means the second, dhj the third d- coefficient of u. dx 1 means the 
square of the d' of x, etc. 

Example 1. The diameter of a sphere increases at a rate of dx = 2-31 
inches per second, when x= 9 5 inches, at what rate (dv=1 does the 

volume v increase? v = n -^- = 0-5 2 3 x 2 . dv = 0-523X3 x~ dx = l -569X2-31 

6 
= 327-1 cubic inches, the answer. 

Example 2. It is found that the displacement of a ship increases as 
x ui the draft of water. At the load draft a = 18 feet the displacement is 
r=2000 tons. Required the displa cement. t = 1) when x = 12 ft. and how 
much (dt. 1) can the vessel be loaded per dx = 1 inch or 1-12 foot, at that 
draft. 

= x^T = 12^X2000 = l0S8<6 t at x = 12 feet dpaft 

a Ub lg J.5 

1-5 Tx^dx V5X2000XY12 „„ nA . . *, 

dt — = — — . — — ] 11-34 tons per inch. 

a 1 -' 3 18 u5 X12in. 

The following page contains the differentials of formulas and trigo- 
nometrical functions. I means the Naperian logarithm. The common 
logarithm log. multiplied by 2-302535 gives the Naperian logarithm I. 





Calculus DiUfere 


nlial Formulas. 


79 


FORMULAS 




DIFFERENTIALS. 


FORMULAS 


DIFFERENTIALS. 


y=x 




dy^dx, 1 


a- 


= e- Z- a dx, 


21 


y=ax* 




dy=2 axdx, 2 


d-l'x 


dx 

X 


22 


y=a?* 




dy=ux ll ~ } dx, 3 


xVx 


= {l-\-l'x)dx, 


23 


Zabx* 


= 


dab x" dx, 4 


l-x 

x n 


(l—l-x)dx, 

x ,l + l 


24 


4 a b" x n 


- 


4 n a b" a"- 1 dx, 5 


X 

Tx 


(l'x — 1 dx 


25 


a-\-x 3 


- 


3 x* dx, 6 


ay 


a y x dx — a a 2 dy 
= -2 J 26 

V / (^+3/ 2 ) 3 


\/x°-\-y- 


(a-ybx'* 


= 


2x(a^-b)dx, 7 


a — 2 b x 
(a^bxf 


2&*a?da? 


27 


Qa b 4 x* — c 


= 


IS a b* x 1 dx, 8 


%/x=x X/ * 


da; 


28 


X-\~Z2P— V 


= 


dx-\-Q z dz—dv, 9 


(ax-\-x") n = n 


[a ar+a; a )«-i(a-J-2 a;)da: 29 


6^ 3 +4a^ 2 ^3.T 


=(18ar-f-8 a x— 3)dx,10 




bxdx 


30 


s /a 2 -{-bx' 2 


x/a^b x 2 


XV** 


= 


V dx-\-2 x v dv, 1 1 


d*(ax*) 


= Gaxdx*, 


31 


xvz = 


(dx .dv ,dz\ 

-- xvz[ — H — 12 

U v ' z J 


d\ax z ) 


= 6adx 3 , 


32 


x(x" — xb") 


= 


(3 a 2 — & 2 x) dx, 13 


d 4 (a x*) 


= obax°-^dx i =o, 


33 


X* 
V 


= 


2xv dx — x" dv 
-„ -» 14 

u 2 


sin.?) 


=+cos.vdv, 


34 


a 

X 


= 


a da; 

— r- 15 

X* 


cas.v 


= — sin.vdv, 


35 


a 

x n 


- 


n a a n - ] dx 
-x— I6 


tan. v 


= + -*_1 

COS. 2 V 


36 


(a+yxy 


= 


Z{a+yxYdff 
2 </x 


cot.u 


dv 

sin. 2 v 


37 


o+^r 


= 


m(a-\r\/x~) m ' l dx 
n f/x 


sec. v 


cos. -ydi* 

C0S. 2 l> 


38 


1 

4(<z — x ) a 


= 


J? 19 

(a— x)»+i 

2adx 

20 

®\/2ax—x? | 


cosec.u 
Tant. for any 


COS. Udl> 

sin. 2 v. 


1 
39 

40 


/ da; 5 * 
curve «=yjl+^i 


2\/2ax—x' 2 

X 



80 Calculus of Integrals. 



The Integral Calculus is the reverse of the Differential, or to find 
the original formula of a given differential. The symbol /is placed be- 
fore the d' to denote that the integral is to be taken out of it, or that the 
original formula is to be found. 

3 a x^-h 1 

The d' of a a? 3 =3 a x" dx, and /3ax"dx = = a ar. 

Rule to find the integral. Add 1 to the exponent of the variable x in 
the d-, divide the d' by the new exponent, dx will disappear, and the 
quotient is the integral. The integral / does not effect a constant. A 
constant term in a formula disappears in its d', consequently any inte- 
gral may have a constant term, whose value is determined by making 
the variable in the integral = o, when the first member in the formula 
will be a constant. It is therefore customary to add a constant C to the 
integral. When it is known that the first member is = o at the same 
time the variable in -the /is — o, then C = o. When a differential is to 
be integrated between two limits of the invariable, say x = a and x = b, 
it is indicated by/ b . or / b 3 c x 1 dx = c(b 3 — a 3 ). 

a a 

Successive differentials are accompanied with the same order of inte- 
grals, as//' 6 x dx" 2, =/3 x" dx = x i . 

The integrals of the differentials gives the formula for the problem. 

Example 3. It is required to find by the calculus a formula for the 
area z of a rightangled triangle. Proposition II, page 78. dz = y dx, the 

formula for the hypothenuse is y =ax, dz=axdx, z=faxdx= IT—IT 1 
or the area z is half the rectangle of the sides x and y. 

Example 4. Find a formula for the convex surface z of a cone, whose 
side is u, and r = radius of the base? Prop. IV, page 78. dz = 2irrdu i 
and z=/2irrdu = Trru, the answer. 

Example 5. Find a formula for the area z of a circle, when it is known 
that the circumference y = 2 n xl Prop. II, page 78. dz = ydx = 2Tr xdx, 
and z —J 2 tt x dx = ir x 2 the answer, x = radius of the circle. 

Example 6. Find a formula for the area z of a parabola of x — abscissa 
or height, and y — ordinate, or half the base 1 Formula for a parabola 

7/ 3 

y = yp x, in which p = the constant parametric diameter, or p = — 

?/ 9 , 2ydy * TT _ l 2y*dy, . /*2y-dy 2y 3 

x = ~ i dx=—^-^- t Prop. II, dz = ydx=— — - and 2= / — - — - = — - = 
p p P J V 3 P 

^-, the answer, or the area of a parabola is % of the base by the height. 

Example 7. Find a formula for the volume v of a paraboloid? Prop. 
,, T , o , 2jry' 3 dy , p2-ny z dy in/ t , y" 1 

III. dv=-ir y" dx = — -, and v=f — - = —^ Li but p = —and v = 

u p ' J p 2p r x 

—y 1 #, the answer. 

V. The center of gravity s from the origin of x, of any plan figure 
bounded by a curved line and rectangular co-ordinates is s = — - 

when z area of the plan. 

VI. The center of gravity s from the origin of x of any solid figure is 
f'x z dx 

s= — j when z = ordinate cross section and v volume of the same. 

v 
Example 8. Find a formula for the centre of gravity (s = 1) of a cone. 

x, when x = height and 

fx z dx 3fTrxy°dx 



The ordinate cross-section z = -mf, and v — — 2/ 2 #, when x = height and 



y radius of the base of the cone. Prop. VI. s == - 

V it y" x 

As the center of gravity is not influenced by the proportion of x and y, 

3 /' 7T X 2 dX 3 7T x^ 

we can make y = x. when s = — — = =- -=% x. from the top. 

7T X 3 4 77 x 3 ' ^ 



Calculus Integral Formulas. 



DIFFERENTIALS. INTEGRALS. 



J±ax z dx — ±afx i dx = ax 4 -\-C, 2 



fx n dx, = 



n-f-1 



+ C, 3 



fy'xdxfx 2 dx = }-C, 4 



/dx /*~ V 

— = /a? /2 d#= 2j / a?4- C, 6 

y^- =fxdx = Z-a? + C, 6 

e/ ar 1 •/ 2 a? ' ' 

/( a re 9 -| )*»=-— +6} / a?+C, 9 

/acta . 
= flZ-a?+C. 10 
x ' 

/•dote 

t/ 0-f-o? 



5Z-(a+a;)+C, 11 



/3 a x li dx 

faxdx-\-3x-dx-b-dx = \-x 3 -b-x-{-C i 

f(a?-\-b-)dx = #(a--}-6 3 H-C, 14 



f(ax—2x 1 fdx = ar 8 (| — aa; -j- ~ )-j- C, 

/3(a x—4?y*(a— 2x(dx=(ax— as 3 ) 3 -}-^ 

rn{x"~^dx) 



DIFFERENTIALS. INTEGRALS. 

dx 



/: 



6 

/3ma; 2 <2a? 



= Z-Or-fya'^-j-^), 21 



m & 3 — m a 3 , 22 



fmxdx = — (& 2 — a 2 ), 23 

a Z 



y' da; 
a 2 +a; s 

o 

/: 






2 a' 



24 

25 



v/^=^" 2 

/=-/ /=/ + / 26 

a 6 a a b 

fsin.xdx =— cos.tf-f-C, 27 

/cos. acta = sin.#-f-Cj 28 

ftsLU.xdx = — Z* cos.a;-f C, 29 

/cot. a;da; = — V sin.ar-f-C, 30 



/i 



sin. a; 
"*dx 



Z'tan. — + C, 31 



\/a-\-x n 

y*2adx 
a 2 ~^x* 



= ya^xn+C, 17 



lUZ+C, 18 

a — a? 



f\ / a I ^x' i dx = — ya 3 +a; 3 -{-— Z-fa; -f 

_^ 2 

/l .'a^hx'dx = — 6(/fl-f-^) 3 +C, 20 



y 1 da; , ^ / t . # \ . ~ 
=Z-tan. -T + — }-\-C,32 
cos. a; \ 4 2 / ' ' 

/ sin.a* cos.a; dx = — sin 3 . x-\-C, 33 

CD 

/sin.&a; , ~ 
da? = 
as 



cos.&a; 



dx 



2 
QO, 



34 



35 



/ 



p dt 

I . ~ = circle arc of which Z=tant. 

/ = circle arc of which 

J \/2x — x* a; = sin.versus. 37 

ff/6adx :i =ff6axdx' 1 =f3ax'>dx=zax a i- C 
fj'2 (a-f b) dx 2 = (a4-&)a; 2 -f- (?, 39 

//2 tf 2 dx 2 -J-8 u x dx du-y-2 xHv s =x i; 3 } 40 



82 Maxima and j>Iiiiima. 



Two variable quantities x and y depended on one another, to find the 
value of one, when the other is a maxima or minima. 

r dx _ 

| is a maxima or minima when its J dy ~ ' 
I first differential coefficient } dy 

d?y 
When the second d- coef. y^ is positive, y is a minimum, and when 

negative y is a maximum. The variables may have both maximums and 
minimums, as formulas will indicate. 
Example l. Find the value of x when y is a maximum or minimum, in 

the formula w = a? 3 — 12 x + 22] dy = (Zx* -,-12) dx. -^ = 3a?— 12 = o. 

dx 

Of which x — V -^ 2 - = 2 the answer. -r~ = 6.r, which is positive, conse- 
quently 2/ = 2 3 — 12 X 2 + 22 = 6, a minimum, when a? =2. 

Example 2. It is required to cut out the strongest possible beam of 
height h and breadth &, from a log of diameter D, fig. 221 page 174] The 
strength of a beam is in proportion to oh? which is "to be a maximum. 

D3= &»+-#», h -=D 2 —b 2 , bh' 2 =b(D°—h 2 ), d(bh?) = (D*—Zb*)db. 

^~^==D-—Zb"-=o, of which the breadth 6-D| JT= 0-577 D, and height 
h= ^/D-— % — D\/ 0-6666 =0-8161 D, the answer. The second d' coef. 

^ - = — 6b, which is negative, and therefore bh 2 is a maximum 
when 6 = 0-577 D. 

Example 3. It is required to know the proportion of heighth h and 
diameter D of a cylinder, having the greatest cubic containt v, with the 

smallest surface '£ including top and bottom? s=— ~-\-TrDh=z— + 7rDft, 

2 
which is to be a minimum. Set v = l and D=l, then z=-r+nh, and 

(4 \ dz 4 nr 

■nr — —)dli, — =7r— — i = 0, when h=- /— =1-12841), the answer. 
hP-J dh li 2 >sj it 

The second d- coef. -^-; = + -£- which is positive, and z a minimum when 

h = 1-1284 D. 

Maclaur in's Theorem* ' 

Maclaurin's Theorem, explains how to develop into a series a function 
with one variable, as 

. % „ x /du\ x' 2 /d.'U\ . X s /d 3 u\ x-* /dm\ 

u =(„) + _(_ ) + _(_J +2x ^_ 3 ;... + ___(_J etc. 

where the factors in the parenthesis is that which it assumes when x=o. 

The function u =-■ — — developed into a series will be 
a-\-x 

_i_ = l--+---...— .etc. 
a -\-x a a 3 ""a 3 a*'"a«+? 

Taylor's Theorem. 

Taylor's Theorem, explains how to develop into a series a function of 
the sum or difference of two variable asw = x-Ly. 

^ {X ^ y) - U ^dx J ^dx* 2 ^'2X3^ dxn 2X3 ...X» 
where u represents the value of the function when y = o. 



Interpolation. 



S3 



Interpolation is to insert numerical values between given datas, for 
constructing tables or empirical formulas expressing the probable rela- 
tive variation of quantities. Let x and y be two variable quantities de- 
pending on one another and measured in simultaneous stages of their 
progress, as 

Xi x 2 x z x f and x 5 
3/1 Hi 3/3 V* and y 5 
We have y — Ay^-rByrrCy^Dy-^Ey-r&Q,. - - - - 1 



2 3 4 

V V V 

[x — x) [x — Xs) [x — x 4 ) (x — ;c ; ) 

~(x\—x 2 ) [Xj— x 3 ) {x l —o: i ) (xHrOk) 

[x~xt) (x — x- 3 ) (x — x<) [x — x,) 



5 given 
Vdatas. 



2> 1 

(X — Xt) (x — x. 3 ) 



-x,) [x — x 5 ) 



bb 3 > 



bfi 



Z>: 



I I 

El) [x— x 2 ) (x—x.) {x—x d ) 



-a 4 > 



I 

{*—*) [x-xj [x — x,) (x-x A ) 



£ {x—x,) [x 6 —x 2 ) [x—x:) [x*-x 4 ) 
5> 



J 



The values of the coefficients A, B, C. P, and E, with their given datas, 
inserted in formula 1 gives an empirical formula for the variation of x 
and y. The number of observations or given datas of x and y should be 
one more than the order of progression. In arithmetical progression 
two observations are sufficient for a correct formula. For all curves in 
the conic sections, or others which are of the second order, there should 
be at least three observations. Pressure of steam progresses with the 
temperature in the 6th order, for which requires seven observations to 
make a correct formula. When the order of progression is not known, 
the more observations gives the most correct result. 

Example. Let y represent the boiling-point of salt water and x the 
percentage of salt in solution. It is found in three experiments, 

that #1= 3, afr=l8, #3=36 per cent. salt. 

when 2fi=213P, 2/2=219°, 2/3=226° boiling-point. 

Find a formula that will give any intermediate value of x and y ? 

Jar— lS)( r-36) £== (^-3)(.r-36) Q= (.?--3)(g— 18) 

(3—13 (3—36)' (18—3)^13—36)' (36— 3) (36— IS)' 



A=- 



jf=213 l-)-219 B-f 226 C. v/=0-40722.c-f211-7 



84 Geometry. 

GEOMETRY. 

DEFINITIONS. 

Demonstration is a course of reasoning by which a truth is established. It 
consists of, 

Thesis, the truth to be established, and, 

Hypothesis, the foundation for the demonstration. 

Axiom is that which is self-evident and requires no demonstration. 

Theorem is something to be proved by demonstration. 

Postulate is something to be done, but is self evident and requires no demon- 
stration. 

Problem is something proposed to be done, and requires demonstration. 

Proposition is either a Theorem or a Problem. 

Corolary is an obvious conseqence deduced from something that has gone 
before. 

Scolium is a remark on preceding propositions, commonly demonstrated *>y 
algebraical formulae. 

Lemma is something premised for a following demonstration. 

Geometrical Quantities* 

Point is a position, but no magnitude. 

A Line is length, without breadth or thickness. 

A Straight Line is the shortest distance between two points. 

Curved line is a length which in every point changes its direction. 

Superficies, Surface, Area, is that which has length and breadth, but n* 
thickness. 

Plane surface is a plane which coincides with a straight line in every direi 
tion. 

Curved surface is a plane which coincides with a curved line. 

Solid has length, breadth and thickness. 

Circle* 

Circle, Cirumference, Periphery, is a curved line drawn on a plane surface, an* 
bounded at a common distance from one point in the plane, (centre.) 

Radius is a line* drawn from the centre in a circle to the periphery. 

Diameter is a line drawn through the centre to the periphery, or the longes* 
line in a circle. 

Chard is any line extending its both ends to the periphery of a circle, and doep 
not go through the centre. 

Arc is a part of a periphery. 

Circle plane, is a plane surface bounded within a circumference. 

Sector is a part of a circle-plane bounded within an arc and two radii. 

Segment is a part of a circle plane bounded within a chord and an arc. 

Zone is a part of a circle included between two parallel chords. 

Lune is the space between the intersecting arcs of two eccentric circles. 

Oval is a round figure having one long and one short diameter at right angles 
to one another. 

Semicircle is a half circle. 

Quadrant is a quarter cf a circle. 

Angles* 

Angle is the opening or inclination of two lines which meet in one point. 
If two radii being drawn from the extremities of a circle arc, to the centre; 
the arc, is a measure of the angle at the centre. 

Right angle is when the opening is a quarter of a circle. 
Acute angle is less than a right angle. 
Obtuse angle is greater than a right angle. 

* Line by itself means a straight line. 



Geometry. 




Triangles* 

Triangle is a figure of three sides. 
Equilateral Triangle, has all its sides equal. 
Isosceles Triangle has two of its sides equal. 
Scalene Triangle has all its sides unequal. 
Bight-angled triangle has one right angle. 
Obtuse-angled, triangle, has one ohtuse angle. 
Acute-angled triangle has all its angles acute. 



Quadrangles* 

Quadrangle is a figure of four sides. 

Parallelogram having its opposite sides parallel, and the opposite angles 
equal. 

Square, having its four sides equal, and four right angles. 

Rectangle, having its opposite sides equal, and four right angles. 

Rhombus, having four equal sides, and opposite angles equal hut not right. 

Rhomboid, same as a parallelogram. 

Trapezium, having four unequal sides. 

Trapezoid, having only two opposite sides parallel. 

Gnomon is the space included between the lines forming two similar parallel- 
ograms, of which the 'smaller is inscribed in the larger, so as to have one com- 
mon angle. 

Polygons* 

Polygons are plane and rightlined figures. 

Regular Polygons are plane figures which inscribe, or circumscribe a circle, 
and their sides being equal. Polygons are named according to their number of 
sides, thus, 

Trigon has three sides. 

Tetragon " four " 

Pentagon " five " 

Hexagon " six '* 

Heptagon " seven " 

For properties of Polygons see page 103. 

Solids* 

Sphere is a solid bounded within a half circle rotating round its diameter. 

Spherical segment, (zone) is a part of a sphere cut off by a plane. 

Spheroid is a sphere Hatted or longed at two opposite sides ; as our earth is 
flatted at the poles, and having one diameter shortest; an egg is longed, and 
having one diameter longest. 

Spindle is a solid bounded within a curved line rotating round its base. 

Cylinder is a solid bounded within a rectangle rotating round one of its sides, 
(axis.) A cylinder has a circle plane^to its base. 

Cone is bounded within a right-angled triangle rotating round one of its sides 
that forms the right angle. 

Ungula is the bottom part of a Cone or Cylinder, cut off by a plane passing ob- 
liquely through the base. 

Cube is bounded within six squares. 

ParalleZopiped is bounded within six parallelograms. 

Prism is a solid described by a rightlined plane moving in a straight line, so 
that the plane forms an angle to its direction line. 

Prismoid is a prism cut oblique!} at the ends. 

Pyramid is bounded between a rightlined plane, and one point at a distance 
from the plane. The sides of the rightlined plane, are bases of triangles deter- 
minating at the aforesaid point, (vertex.) 

Perimeter is the sum of all the sides in a figure, plane or solid. 

Polyhedrons. See page 95, for properties and names of the five regular poly- 
hedrons. 



Octagon 


has eight 


sides. 


Nonagon 


tt 


nine 


" 


Decagon 


tt 


ten 


« 


Undecagon 


i( 


eleven 


tt 


Dodecagon 


ii 


twelve 


tt 



Constructions. 






37. 

To construct an ellipse. 

With o as a centre, draw two concentric 
circles with diameters equal to the long and 
short axes of the desired ellipse. Draw from 
o any number of radii, A, B, &c. Draw the 
line B b' parallel to n and b b' parallel to w, 
then b' is a point in the desired ellipse. 



38. To draw an ellipse with a string. 

Having given the two axes, set off from c 
half the great axis at a and b, which are the 
two focuses in the ellipse. Take an endless 
string as long as the three sides in the triangle 
o, b, c, fix two pins or nails in the focuses one 
in a, and one in b, lay the string round a, and 
b f stretch it with a pencil d, which then will 
describe the desired ellipse. 



39. 



To draw an ellipse by circle arcs. 





Divide the long axis into three equal parts, 
draw the two circles and where they intersect 
one another are the centres for the tangent 
arcs of the ellipse as shown by the figure. 



40. To draw am ellipse by circle arcs. 

Given the two axes, set off the short axis 
from A to b, divide b B into three equal parts, 
set off two of these parts from o towards c 
and c which are the centres for the ends of 
the ellipse. Make equilateral triangles on c 
c, when e e will be the centres for the sides of 
the ellipse. If the long axis is more than 
twice the short one, this construction will not 
make a good ellipse. 



41. 



To construct an ellipse. 



Given the two axes, set off half the long axis 
from c to//', which will be the two focuses in 
the ellipse. Divide the long axis into any num- 
ber of parts, say a to be a division point. Take 
A a as radius and / as centre and describe a 
circle arc about b, take a B as radius and / as 
centre describe another circle arc about 6, then 
the intersection b is a point in the ellipse, and 
so the whole ellipse can be constructed. 



42. 

To draw an ellipse that will tangent two 
parallel lines in A and B. 

Draw a semicircle on A B, draw ordinates 
in the circle at right angle to A B, the corre- 
sponding and equal ordinates for the ellipse 
to be drawn parallel to the lines, and thus the 
elliptic curve is obtained as shown by the 
figure. 



Constructions. 



87 




43. To construct a cycloid. 

The circumference C=3-14 D. Divide the 
rolling circle and base line C into a number 
of equal parts, draw through the division 
point the ordinates and abscissas, make a a' 
= 1 d,b & '=2' e, c c =3'/, then a' b' and d are 
points in the cycloid. In the Epicycloid, and 
Hypocycloid the abscissas are circles and the 
ordinates are radii to one common centre. 



44, 



Evolute of a circle. 



Given the pitch p, the angle v, and radius r. 
Divide the angle v into a number of equal 
parts, draw the radii and tangents for each 
part, divide the pitch p into an equal number 
of equal parts, then the first tangent will be 
one part, second two parts, third three parts, 
&c, and so the Evolute is traced. 



45. To construct a spiral with compasses 
and four centres. 

Given the pitch of the spiral, construct a 
square about the centre, with the four sides 
together equal to the pitch. Prolong the 
sides in one direction as shown by the figure, 
the corners are the centres for each arc of the 
external angles. 



A / a 




46. To construct a Parabola. 

Given the vertex A, axis x, and a point P. 
Draw A B at right angle to x, and B P parallel 
to a?, divide A B and B P into an equal num- 
ber of equal parts. From the vertex A draw 
lines to the divisions on B P, from the divi- 
sions onAB draw the ordinates parallel to x, 
the corresponding intersections are points in 
the parabola. 




47. To construct a Parabola. 

Given the axis of ordinate P, and vertex A. 
Take A as a centre and describe a semicircle 
from B which gives the focus of the parabola at 
/. Draw any ordinate y at right angle to the 
abscissa A x, take a as radius and the focus/ 
as a centre, then intersect the ordinate y, by 
a circle-arc in P which will be a point in the 
parabola. In the same manner the whole 
Parabola is constructed. 



48. 




To draw an arithmetic spiral. 
Given the pitch p and angle v, divide them 
into an equal number of equal parts say 6. 
make 01=01, 2=02, 3=0 3, 4=04, 05=0 5, 
and 6=the pitch p; then join the points 1, 2, 
3, 4, 5, and 6, which will form the spiral re- 
quired. 



Circle. 



CIRCLE. 

The periphery of a circle is divided into 360° (degrees) equal parts, each called 
a degree. ' 

One degree = 60' (minutes.) 
One minute = 60" (seconds.) 
Half a circle (hemisphere) = 180°. 
Quarter of a circle (quadrant) = 90°. 

By the accompanying formula any part of the circle can be calculated. 
'Formula for tlie Circle* 



p = Trd = 314d, - - 

p = 2*r = 6-28r, - - 
p = 2}fea = 3-54]/a7 



3-14' 



l=s \/^= 1 ' m V"> ' 



— P _ P 
r ~Z"~ 6-28 



7t 
16406: 

2* 

3*- 
4*- 
byr 



1, 

■2, 

6, 



= 0-564^a, 



a = — = 0-785J2 

4 ' 

a = jrr2 = 3-14T 2 , 

a "~ 4a- "~ 12-56' 



«- 2 ~ 4* • * * * 

a = — = l.b7rd, - - • 



10, 
11, 
12. 



3-1415926535897932384626433832795028841971693993751058209749445923078 
2862089986280348253421170679821480865132823066470938446 



7sr 
8?r 
9?r 

Ye* 



ico" 



= 6-28218530710000. 
= 9-42477796070000. 
= 12-5663706143000. 
= 15-7079632679000. 
= 18-8495559215000. 
= 21-9911485751000. 
= 25-1327412S870OO. 
= 28-2743338823000. 
= 1-57079632679000. 
= 0-78539816339700. 
= 1-04719755119600. 
= 0-52359877559800. 
= 0-39269908169800. 
=• 0-26179938779900. 
— 0-00872667621060. 



i. = 0-31830988618370. 
1 =0-63661977236740. 



r = radius of the circle. 

d= diameter. 

p = periphery. 

a = area of a circle, or part thereof. 

b = circle-arc, length of. 



- =1-27323954473480. 

- = 0-95492965855110. 

7T 

- = 1-90965931710220. 
n 

- = 3-81971863420440. 

360 

— = 114-591559026122. 

r 2 = 9-86965000000000. 

^ = 1-77245300000000. 

V i- = 0-56418900000000. 



Letters denote, 

c = chord of a segment, length o£ 

ft — height of a segment. 

s = side of a regular polygon. 

v = centre angle. 

w = polygon angle. 



j^Be careful to express all the dimensions by the same unit.as miles, rods, 
yards, feet, or inches, &c, &c, or else the calculation will be wrong. 

Example 1. Fig. 49. The diameter of a circle is 8 feet, 8 inches, how long is the 
circumference ? 

Formula 1. p = ^d = 314X8-666 = 27-211 feet, the answer. 



LONGEMETRT. 



89 




49. 

The periphery of a Circle is commonly expressed 
by the Greek letter ar = 3*14 when the diameter 
d = 1 or the unit. For any other value of the di- 
ameter d, we will denote the periphery by the let- 
ter p, r = radius, and a = area of the circle. The 
periphery of a circle is equal to 32i times its diam- 
etag, c = chord. mo 



50. 




Ttr r 



51. 




w -180-^, 

t; - 2(180° — w). 



52, 



r = 



-gr"~""2A' 



c = 2 sJZhr — A 2 . 




53. 



AC 






54. 



/ 2 ,<*+& — #K% 
b\/*-{— 21— ) 



90 



Longemetht. 





D/JC cVJL 





V = V, w==w a 

w + v = 180°, w > v. 

f 



56. 



D = B + C, A' + B' + C = 180°, 
B = D-C, A+ B+ C=180°, 



57. 



A + J5 + C « 180°, 
4' -4, 5' = £. 



58, 



JS+C-^l + D-180 , 
D - 5 + c, 
JS = J. + B. 



f a, y b v 




7 
. 9 " 

6 


^ 


ab 



59. 



(a + a) a «a*+2a5 + $ a . 



60. 



a^ 



(cu^y 



^*t 



¥*■ 



/\ 



(a-Z0 a = a*-2a$ + &\ 



LONGEMETRY. 



91 






61. 







Zm 






62. 



a 2 £ = c : d> 
ad = #c, 



63, 



a : £ = c : d t 

ad =» be. 



U, 



65. 



66. 



a : e =» c : b 9 

c — yfab. 



AlB-ail. 



a : x — x : a ~ x, 



s/^WM 



92 



LONGEMETRT. 





67. 



c* - a* + b\ 
a* = c* - 6% 
#» = c a - a\ 



68. 



c* ~ a* H-5 a - 2bd, 



dJ 



25 




69. 



c* = a* + 5* + 25 d, 
c* - a* - 5* 



<* 



25 



70. 




a : b = A : c, 
ere oi 

"" 5~~ c . 
j c a eh 



71. 




a : c = d : (5 — d) 9 

ab 
c + a 



d=. 




72. 



a : c = 5 : d, 

ad =* 5c. 



LONGEMETRY. 



93 




73, 



a : t - t : b, 




74. 



*•-(<! + J) (a -6), 



<t> — * 



75. 




t-Stf-p 



aR 



= J2 — , «-Vf+(12-r)% 



* - V a* - (-K - r)% sin.u = -• 



0^© 



76, 







77. 







78. 



J - n V rc 2 d a + P 3 > 
2 



?* = 



V^ 3 ^ + P." 



04 



LONGEMETRT. 




• 9- To find the length of a Spiral. 



7 Ttr* 

I = nr n = — , 



n = 



nr P 



P=«L*=L. p- Pitch. 
I n 




80. To find the length of a Spiral. 
l = rt7i(R+r), 



*- 5 (*■—). 



81. 



Periphery of an Ellipse. 




p = 2 \/2> a + 1-4674^. 




82. 



To construct a screw IM/r. 




83. To square a Circumference. 

R - 0-556355 d = 1-1127 r - 0-7071 & 
S = 0-785398 d = 1-57079 r - 1-4142 22. 
cZ = 1-27322 S = 1-79740 R = 2 r. 




84. To square a Circleplane. 

R = 0-650744 rf - 1-30148 r =0-7071 5. 
5 -* 0-886226 d = 1-77245 r = 1-4142 R. 
d = 1-12831 S = 1-5367 £ = 2 r. 



Polyhedrons. 



P5 





85. Tetrahedron, 

r = 0*20413 s. 
R = 0-60725 s. 
a = 1-73205 s 3 . 
c = 0-11785 S3. 



86. 



Hexahedron, 

r = 0-50000 g. 
i? = 0-86602 s. 
a = 6-00000 s». 
c = 1-00000 S'. 




87. 



Octahedron, 

r = 0-40721 s. 
R = 0-71710 s. 
a = 3-46410 s*. 
c = 0-47140 s» 




88. Dodecahedron, 

r = 1-11350 s. 
R= 1-36428 s. 
a = 20-5457 s>. 
c = 7-66312 s«. 




89. 



Icosahedron, 

r = 0-7558 s. 

£= 0-9510 s. 
a = 8-66025 s 3 . 
c = 2-18169 s a . 



r = Kadius of an inscribed Sphere. 
R = Radius of circumscribed Sphere. 
a = Area of the Polyhedrons. 
c = Cubic contents of the Polyhedrons. 
S = Side or edge of the Polyhedrons. 



06 



Planemetry. 



f — 
Wmm 



90. Square. 

a = g* = U\ 

a = 0-7071<P = 2-8284 c\ 



WW' 1 * 




91. 



Rectangle. 
a = a b 9 
a = b J d*-b* 



92. 



Triangle. 



— W'-e^" 




93. Triangle 

a =- i&A, 
3 



a 



-2\A-c 



c* - a* - frV 



25 



)! 




94. 



Quadrangle, 



a = 4^(a + 5). 




95. Quadrangle. 



a=l(a[h+h'l + 6£' + cA). 



Planemetry. 



97 






9C. Circle Plane. 

a = tfr* = 0-785<2 8 , 
a = |^= 0.0794 P a . 



97, 



Circle Ring. 



a = tf (# a - r 2 ) = n(R + r){R-r) t 
a=0-785(D*-<T). 



98. 



Sector. 

a = i5r, 

!75f r 2 v r* v 



360 "114-5 




99. Segment. 

a = J.[Jr — c (r — £)], 



360 +2 



i^:» 




100. Quadrant. 



a = 0-785 r a = 0-3916 c\ 




101. 



a = 0-215 r 3 = 0-1075 e*. 



98 



PlANEMETRY. 




102. Ellipse. 



a = 7tRr= 0-785 D d. 




103. Parabola. 







a = 3 hvph. 



104. Irregular Figure. 



"a =6 (A + /*' + £"). 



105. 



Ellipsoid. 



a = 8-88r VR* +r a , 



a = 2-22 d VD* + d a . 



106. 



Cylinder. 

a. = 2nrh = ndh, 

a a 



fc = 



2/rr it d 



Tlic road to Extremity of Space. 

1st. Draw a Circle and inscribe a Square, and in 

that Square a Circle, &c, &c, &c The last 

figure that can be drawn, is one extremity of Space 
Required if the last one is a Circle or a Square? 

2nd. Draw a Circle and circumscribe a Square, 

and around that Square a Circle, &c, &c, &c 

The last one that can be circumscribed is the other 
extremity of space. Required if the last figure is 
a Circle or a Square ? 




Surface op Solids, 

107. 



99 



Sphere, 
a = 4 Tir* = 12-56 r a = n d\ 





108. Forus. 

a =4^ a £r = 39-44 Rr, 
a = 9-86 D d. 



109. Sphere fifccfor. 



a=-^(4A + c). 




110 Circle Zone. 




111. Cone. 

&,= 7t R s, 
a=7iR sTrFTT*. 




112. 



co = 
a = 

u = 



7T* 



Cone. 



R=s + 



d s 

iDZTd' 



2 v-D + rf), « 
180 J 180(P-<Q . 



100 



Stereometry. 



113. 




Sphere. 



4 7t r 3 

c = -J^-= 4-189 r% 
c-^iL- 0-523 <P. 






114. Fon/s. 

C = 2tfa.Rr 2 = 19-72 #r a , 
c= 2-463 Dd*. 



115. 



Sphere Sector. 
hitr 1 h = 2-0944 r* h, 



C = **i*(r + V^-ic 3 ). 



116. Zone. 



C a + 4h* 




117. 



Cone. 



71 r 1 h 

C= = 1-046 r'A, 



c = 0-2616 <T A. 






£- ; 



•P : :; : ^'^ 



118. Come Frusirum. 

c = hrt MR* + Rr + r 9 ), 



9 

Stereometry. 



101 




119. 



Cylinder. 
c^rtr* A -0.785 d* h, 



c = 



'_= 00796 ;>* h. 



R^^ 





< 7* > 



120. Ellipsoid. 

C = 0-424 n*Rr* = 4-1847 R r 8 , 
c = 0053 ^ D d* = 0-5231 D d\ 



121. Paraboloid. 



c = brtr*h = 1-5707 r a A. 




122. Pyramid. 

C = 4 a A, 



6 V r 





123. Pyramidic Frustrum. 



c = q04 + a + V^la). 



124. Wedg-e Frustrum. 

hs, , x 
C = -2"(a + fy 



102 



Stereometry. 






'■hi* > 



125. Cask. 

c = 1-0453 2(0-4 X> 3 + 0-2 D d + 0-15^), 

Gallon = o^n(4D' 2 + 2 D<* + 1- W*)- 



2200 



126. Cylinder Sections. 

C = rtr 2 (Z + Z' _|r), 
© « * r*(l + Z') - 2-1 r\ 



127. Circular Spindle. 

c = ar(J c 3 - 0-2 rf[c+| V^+d 5 ] vQ^Tc 5 ) 



I 



Example 1. Fig. 92. The base of a Triangle is 5 = 8 feet, 3 inches, and the 
height, h = 5 feet, 6 inches. What is the area a = ? 



b h 8-25 X 5-5 



= 22-6875 square feet. 



2 ~~ 2 

Example 2. Fig. 9S. A Circle Sector having an angle v = 39° and the radius 
r = 67 1 inches. What is the area of the sector a = ? 
_ 7lr* v _ 3*14 X 67-75* X 39° 



360 



360 



= 1562-1 square feet. 



Example 3. Fig. 110. A Spherical Zone having its diameter c = 18£ inchea 
and height h = 7f inches. What is the convex surface of the Zone ? 

a=== 2?(c* + &9) =~i /18-59 + 7'75») = 315-96 square inches. 

Example 4. Fig. 88. Require the radius R of a Sphere that will circumscribe 
a Dodecahedron with the side s = 9 inches. 

R = 1-36428 X 9 = 12-27852 inches, the answer. 

Example 5. Fig. 118. A Frustrum of a Cone having its bottom diameter D = 13 
inches, the top diameter d = 5£ inches, and the height h = 25 inches. What is 
the cubic contents e = 1 

c =, T V n h(D* + D d + d*)=-- 0-2618 X 25 (l3* + 13 X 5*25 + 525*)= 20995 
cubic inches . 

Example 6. Fig. 125. A Cask having its bung diameter D = 36 inches, head 
diameter d = 28 inches, and length l — 56 inches, (inside measurement) how 
many gallons of liquid can be contained in the cask ? (The gallon = 231 cub. in.) 

Gallon = J^(4 X 20* + 2 X 36 X 23 + 1-5 X 28*)= 214 gallons. 



Geometry. — Table or Polygons. 



103 



Example 7. Fig. 50. Require the length of the circle-arc b, when the angle 
v = 42°, and the radius r = 4 feet, 3 inclies? 






3-14X4- 25X42 
ISO 



= 3*113 feet. 



Example 8. Fig. 52. Require the radius of a circle-arc, whose chord is 9 feet, 
4 inches, and height, h = 1 foot, 8 inches ? 



c*+4fc2 9-332-J-4X1-66* 9S-0711 



SXl'66 



13*28 



= 7*3S4 feet. 



Example 9. Fig. 
and c = 8-66 feet. 



8. The three sides in a triangle "being, a - 
How high is the triangle oYer the base b 1 



3-42, 6 = 7-75, 



a 2 + -f,a_o2 6-422-] 



26 



2XS-66 



= 1-5175 feet, 



the height h --= j/a- — da = ^6.42* — 1*517S* = 6-24 feet, the answer. 

Example 10. Fig. 77. The radius of a walking beam is, r = 8*36 feet, the stroke 
5 = 5-5 feet. How much is the vibration V=1 



Vibration, 



'=r— a / r a — £1 = 8-36 — \ / 8-363 _J 
=0-471 feet = 5*65 inches = 5"—, the answer. 



TABLE OF POLYGONS. 



5_^2 
4 



Number 

of sides 

in the 

Polygon. 



Trigon. 

Tetragon. 

Pentagon. 

Hexagon. 

Heptagon. 

Octagon. 

Nonagon. 

Decagon. 

Undecagon. 

Dodecagon. 



120° 

90° 

72° 

60° 

51°43' 

45° 

40° 

36° 

32°13' 

30° 

25°43 / 

24° 

22°30' 

20° 

18° 



Polygon 
Angle v. 



24 ! 15° 



60° 

90° 
108° 
120° 
'.128°17' 
135° 
140° 
|144° 
147 °47' 
jl50° 
154°17' 
1156° 
157°30' 
160° 
162° 
165° 





Sid8 


Area 


A pot em 


Side 




= kB. 


= *s». 


= hR. 


= k r. 






(S& 


rfS 


A 




^%J 


MS/ 


<^y 


XX 




* ir -i 


1-732 


0-4330 


0-5000 


3-4641 


1-4142 


1-0000 


0-7071 


2-0000 


1-1755 


1-7205 


0-8090 


1-4536 


1-0000 


2-59S0 


0-8660 


1-1547 


0-8677 


3-6339 


0-9009 


0-9631 


0-7653 


4-82S4 


0-9238 


0-8284 


0-6840 


6-1820 


0-9396 


0-7279 


0-61S0 


7-6942 


0-9510 


0-6493 


0-5634 


9-3656 


0-9595 


0-5872 


0-5176 


11-196 


0-9659 


0-5359 


0-4450 


15-334 


0-9762 


0-4562 


0-4158 


17-642 


0-9781 


0-^250 


0-3900 


20-128 


0-9S07 


0-4068 


0-3472 


125-534 


0-9848 


0-3526 


0-3130 140-634 


0-9S77 


0-3166 


» 0-2610 45-593 


0-9914 


0-2632 



5-1961 
4-0000 
3-6327 
3-4640 
3-3710 
3-3130 
3-2750 
3-2490 
3-2290 
3-2152 
3-1935 
3-1882 
3-1824 
3-1737 
3-1676 
3-1596 



Explanation of the Table for Polygons* 

The number of sides in the polygon is noted in the first column. 
Jc = tabular inefficient, to be multiplied as noted on the top of the columns. 
Example 1. How long is the side of an inscribed Pentagon, when the radius 
of the circle is 3 feet, and 4 inches? (4 inches = 0-333 feet.) 

3-333X1*1755 = 3-9179 feet, the answer. 
Example 2. "What is the area of a Heptagon when one of its sides is 13*75 inches 

13-752X3-0339=6S7-02 square inches. 



104 



Circumferences and Areas of Circles. 



I 


Circ. 1 


Diame- f \, 


tar. ^ J 


1 


•0981 


V* - 


•1963 


£ ~ 


•3926 


3 

1 «? _ 


•5890 


I — 


•7854 


1% - 


•9817 


t .- 


1-178 


7 
IS" - 


1-374 


i_ 


1-570 


A - 


1-767 


£ - 


1-963 


« - 


2-159 


1 — 


2-356 


M - 


2-552 


* - 


2.748 


il - 


2-945 


x 


3-141 


- 


3-534 


i r" 


3-927 


4-319 


i-- 


4-712 




5-105 


2 -- 


5-497 




5-890 


a — 


6-283 




6-675 


i -'- 


7-068 




7-461 


i- 


7-854 




8-246 


f -- 


8-639 


3 


'9-032 


9-424 




9-817 


i - 


10-21 




10-60 


*-- 


10-99 




11-38 


1 -- 


11-78 




12-17 


4 — 


12-56 




12-95 


i - 


13-35 




13-74 


i- 


14-13 




14-52 


2 - 


■ 14-92 




■ 15.31 


J 


- 



Area. 



•1104 

•1503 

•1963 

•2485 

•3067 

•3712 

•4417 

•5184 

•6013 

•6902 

•7854 

•9940 

1-227 

1-484 

1-767 

2-073 

2-405 

2-761 

3-141 

3-546 

3-976 

4-430 

4-908 

5-411 

5-939 

6-491 

7-068 

7-669 

8-295 

8-946 

9-621 

10-320 

11-044 

11-793 

12-566 

13-364 

14-186 

15-033 

15-904 

16-800 

17-720 

18-665 



Circ. 



Diame- 
ter. 



7- 

i 
i- 

I 
8- 

1 
I- 
I 

9- 
l 
h- 
I 
10- 

i 
I- 



15-70 
16-10 
16-49 
16-88 
17-27 
17-67 
18-06 
18-45 
18-84 
19-24 
19-63 
20-02 
20-42 
20-81 
21-20 
21-57 
21-99 
22-38 
22-77 
-J23-16 
23-56 
23-95 
24-34 
24-74 
25-13 
25-52 
25-91 
26-31 
26-70 
27-09 
27-48 
27-88 
28-27 
28-66 
29-05 
29-45 
29-84 
30-23 
30-63 
31-02 
31-41 
31-80 
32-20 
32-59 
32-98 
33-37 
33-77 
34-16 



Area 



19-635 
20-629 
21-647 
22-690 
23-758 
24-850 
25-967 
27-108 
28-274 
29-464 
30-679 
31-919 
33-183 
34-471 
35-784 
37-122 
38-484 
39-871 
41-282 
42-718 
44-178 
45-663 
47-173 
48-707 
50-265 
51-848 
53-456 
55-088 
56-745 
58-426 
60-132 
61-862 
63-617 
65-396' 
67-200 
69-029 
70-882 
72-759 
74-662 
76-588 
78-539 
80-515 
82-516 
84-540 
86-590 
88-664 
90-762 
92-885 



Diame- 
ter. 



Circ. 

o: 



Area- 



11 -r 


34.55 




34-95 


i - 


35-34 




35-73 


h~ 


36-12 




36-52 


1 - 


36-91 




37-30 


12 — 


37-69 




38-09 


i - 


3S-48 




38-87 


\" 


39-27 




39-66 ! 


l - 


40-05 




40-44 


13- 


40-84 




41-23 


l - 


41-62 




42-01 


l~ 


42-41 




42-80 


2 - 


43-19 




43-58 


14- 


-43-98 




44-37 


i - 


- 44-76 




-4546 


*- 


-45-55 




- 45-94 


f - 


- 46-33 




- 46-73 ! 


15— 


-47-12 




- 47-51 


I - 


- 47*90 




- 48-30 


J- 


- 48-69 




-49-08 


l - 


- 49-48 




- 49-87 


16- 


- 50-26 




- 50-65 


i - 


- 51-05 




- 51-44 


I- 


- 51-83 




• 52-22 


1 - 


- 52-62 




- 53-01 







CIRCUMFERENCES 


and Areas of Circles. 




105 


Circ. 


Area. 


Circ. Area 


Circ. 1 Area. 


Diame- / \ 


/fj^ 


Diame- ( \ 1 


Diame- f \ ! /C s 


te, ^ 


i|JP 


te, ^J 


\£P" 


ter. \^J Hp 


ir-j 


53-40 


226-98 


23-j 


72-25 


415-47 


29- 


91-10 ! 660-52 




53-79 


230-33 




72*64 


420-00 




-91-49 ,666-22 


i - 


54-19 


233-70 


i - 


73-04 


424-55 


1 - 


91-89 i 671-95 




54*58 


237-10 




73-43 


429-13 




-92*28 ; 677-71 


i— 


54-97 


240-52 


I— 


73-82 


433-73 


i— 


92-67 | 683-49 




55-37 


243-97 




74-21 


43S-30 




93*06 j 689-29 


i ~ 


55-76 


247-45 


1 - 


74-61 


443-01 


t - 


93-46 695-12 




56-16 


250-94 




75- 


447-69 




93-85 


700-98 


13- 


56-54 


254-46 


24— 


75-39 


452-39 


30— 


94-24 


706-86 




56-94 


258-01 




75-79 


457-11 




94-64 


712-76 


i - 


57-33 


261-58 


I ~ 


76-18 


461-86 


i " 


95-03 


71S-69 




57-72 


265-18 




76-57 


466-63 




95-42 


724-64 


£"~ 


58-11 


268-80 


V~\ 


76-96 


471-43 


*— 


95-81 


730-61 




58-51 


272-44 




77-36 


476-25 




96-21 


736-61 


1 - 


58-90 


276-11 


1 - 


77-75 


481-10 


f - 


96-60 


742-64 




- 59-29 


279-81 




78-14 


4S5-97 




96-99 


748-69 


19- 


-'59-69 


283-52 


25—- 


78-54 


490-87 


31- 


97-38 


754-76 




^60*08 


287-27 




78-93 


495-79 




97-78 


760-86 


i - 


-60-47 


291-03 


i " 


79-32 


500-74 


l - 


98-17 


766-99 




-'60-86 


294-83 




79-71 


505-71 




98-56 


773-14 


\- 


-61-26 


298-64 


J~ 


-80-10 


510-70 


l~ 


98-96 


779-31 




61-65 


302-48 




-80-50 


520-70 




99-35 


785-51 


i - 


62-04 


306-35 


1 - 


-80-89 


f ~ 


99-74 


791-73 




62-43 


310-24 




-81-28 


525-83 




100-1 


797*97 


20- 


-62-83 


314-16 


26 ~~ 


-81-68 


530-93 


32- 


100-5 


804-24 




^63-22 


318-09 




-82-07 


536-04 




100-9 


810-54 


l - 


-' 63-61 


322-06 


l- 


-82-46 


541-18 


l - 


101-3 816-86 




-'64-01 


326-05 




-82-85 


546-35 




101-7 1823-21 


\- 


- 1 64.40 


330-06 


\- 


-83-25 


551-54 


l~ 


102-1 829-57 




-,64-79 


334-10 




-83-64 


556-76 




102-4 835-97 


I - 


J 65-18 


338-16 


i- 


-84-03 


562-00 


I - 


-102-8 1 842-39 




-65-58 


342-25 




-84-43 


567*20 




103*2 848*83 


21- 


-65-97 


346-36 


27- 


- 84-82 


572*55 


33— 


103*6 855*30 




-166-36 


350-49 




-85-21 


577-87 




104- 861-79 


l - 


-i66*75 


354-65 


i- 


-85-60 


583-20 


i - 


104*4 868-30 




- 67-15 


358-84 




-86- 


588-57 




- 104-8 874-84 


I- 


-67-54 


363-05 


h- 


-86-39 


593-95 


*- 


105-2 I 881-41 




-67-93 


367-28 




-86-78 


599*37 




-105-6 j S8S-00 


I - 


-68-32 


371-54 


1 - 


-87*17 


604-80 


I - 


-106- 894-61 




- 1 68-72 


375-82 




-87-57 


610-26 




-106-4 901-25 


22- 


{69-11 


380-13 


2S— 


-87-96 


615-75 


34— 


-106-8 1907-92 




-69-50 


384-46 




-88-35 


621-26 




-107-2 ! 914-61 


l - 


-69-90 


388-82 


I - 


-88*75 


626*79 


I ~ 


-107-5 921-32 




-^70-29 


393-20 




-89-L4 


632-35 




■107-9 928*06 


\- 


^70*68 


397-60 


J- 


-89-53 


637-94 


I- 


- 108-3 934-82 




^71-07 


402-03 




-89-92 


643-54 




-108-7 1941-60 


l - 


i71-47 


406-49 


s - 


- 90-32 


649-18 


l ' 


- 109*1 948-41 




-71-86 


410-97 




-90-71 


654-83 




■109-5 j 955-25 






j 


J 1 




• 



106 




Circumferences a^td areas op Circles. 






Circ. 


Area. 


Circ. 1 Area. 


Circ. | Area, 


Diame- f \ 
tor. ^J 


IP 


Diame- f \ 
te, {J 


w 


Diame- f \ 
te, ^J 


m 


35 -r 


■,109-9 


962-11 


41-i 


128.8 


1320-2 


47- 


-,147-6 


1734-9 




110-3 


968-99 




129-1 


1328-3 




148- 


1744-1 


i- 


110-7 


975-90 


I - 


- 129-5 


11336-4 


l - 


- 148-4 


1753-4 




111-1 


982-84 




129-9 


1344-5 




- 148-8 


1762-7 


*- 


111-5 


989-80 


*- 


130-3 


1352-6 


\- 


- 149-2 


1772-0 




111-9 


996-78 




130-7 


1360-8 




- 149-6 


1781-3 


1- 


112-3 


1003-7 


i - 


131-1 


1369-0 


l - 


-150- 


1790-7 




112-7 


1010-8 




131-5 


1377-2 




- 150-4 


1800-1 


36- 


113- 


1017-8 


42— 


131-9 


1385-4 


48— 


-150-7 


1809-5 




113-4 


1024-9 




132-3 


1393-7 




-150-1 


1818-9 


£ - 


- 113-8 


1032-0 


i " 


132-7 


1401-9 


I ~ 


151-5 


1828-4 




114-2 


1039-1 




133-1 


1410-2 




- 151-9 


1837-9 


2~~ 


114-6 


1046-3 


h~ 


133-5 


1418-6 


\' 


- 152-3 


1847-4 




115- 


1053-5 




133-9 


1426-9 




- 152-7 


1856-9 


i - 


115-4 


1060-7 


1 - 


134-3 


1435-3 


l 


- 153-1 


1866-5 




- 115-8 


1067-9 




134-6 


1443-7 




- 153-5 


1876-1 


37- 


- 116-2 


1075-2 


43- 


135- 


1452-2 


49- 


153-9 


1885-7 




-116-6 


1082-4 




135-4 


1460-6 




- 154-3 


1895-3 


i - 


-117- 


1089-7 


i ' 


135-8 


1469-1 


l - 


154-7 


1905-0 




-117-4 


1097-1 




136-2 


1477-6 




- 155-1 


1914-7 


*- 


- 117-8 


1104-4 


*- 


136-6 


1486-1 


h~ 


155-5 


1924-4 




-118-2 


1111-8 




137- 


1494-7 




- 155-9 


1934-1 


2 - 


-118-6 


1119-2 


I 4 - 


137-4 


1503-3 


l ~ 


156-2 


1943-9 




-118-9 


1126-6 




137-8 


1511-9 




156-6 


1953-6 


38- 


- 119-3 


1134-1 


44-- 


138-2 


1520-5 


50- 


157- 


1963-5 




119-7 


1141-5 




138-6 


1529-1 




157-4 


1973-3 


l - 


-120-1 


1149-0 


*-~ 


139- 


1537-8 


il 


157-8 


1983-1 




-120-5 


1156-6 




139-4 


1546-5 




158-2 


1993-0 


*- 


-120-9 


1164-1 


£*— 


139-8 


1555-2 


*- 


158-6 


2002-9 




-121-3 


1171-7 




140-1 


1564-0 




159- 


2012-8 


1 - 


-121-7 


1179-3 


1- 


140-5 


1572-8 


i - 


159-4 


2022-8 




122-1 


1186-9 




140-9 


1581-6 




159-8 


2032-8 


39- 


122-5 


1194-5 


45-- 


141-3 


1590-4 


51-- 


160-2 


2042-8' 




122-9 


1202-2 




141-7 


1599-2 




160-6 


2052-8 


l - 


123-3 


1209-9 


l - 


142-1 


1608-1 


i - 


161- 


2062-9 




123-7 


1217-6 




142-5 


1617-0 




161-3 


2072-9 


*- 


124- 


1225-4 


k~ 


142-9 


1625-9 


1- 


161-7 


2083-0 




- 124-4 


1233-1 




143-3 


1634-9 




162-1 


2093-2 


1 '- 


124-8 


1240-9 


s - 


143-7 


1643-8 


1 - 


162-5 


2103-3 




125-2 


1248-7 




144-1 


1652-8 




162-9 


2113-5 


40- 


-125-6 


1256-6 


46 — 


144-5 


1661-9 


52-- 


163-3 


2123-7 




-126- 


1264-5 




144-9 


1670-9 




163-7 | 


2133-9 


i - 


126-4 


1272-3 


i - 


145-2 


1680-0 


l - 


164-1 


2144-1 




126-8 


1280-3 




145-6 


1689-1 




164-5 2154-4 


i- 


127-2 


1288-2 


£-- 


146- 


1698-2 


*— 


164-9 2164-7 




127-6 


1296-2 




146-4 


1707-3 




165-3 2175-0 


£ ■ 


128- 


1304-2 


2 -- 


146-8 


1716-5 


1 " 


165-7 21S5-4 




128-4 


1312-2 




147-2 


1725-7 




166-1 2195-7 












l 







Circumferences 


and Areas of Circles. 




107 


Circ. 


Area. 


Circ. 


Area. 


Circ. Area. 


Diame- / ^ 


$ 


Diame- ( \ 


j§Hfei 


Diame- f \ 0m0\ 


ter. ^ ) 


ter. ^ J 




ter. 1 J |p j 


53-y 


166-5 


2206-1 


59-j 


185-3 


2733-9 


65- 


204-2 


3318-3 




166-8 


2216-6 




-185-7 


2745-5 




-204-5 


3331-0 


l- 


167-2 


2227-0 


i " 


186-1 


2757-1 


* - 


204-9 


3343-8 




167-6 


2237*5 




186-5 


2768-8 




205-3 


3356-7 


I-- 


168- 


2248-0 


J- 


186-9 


2780-5 


i- 


205-7 


3369-5 




168-4 


2258-5 




187-3 


2792-2 




206-1 


3382-4 


1 - 


168-8 


2269-0 


i - 


187-7 


2803-9 


1 - 


206-5 


3395-3 




169-2 


2279-6 




188-1 


2815-6 




206-9 


3408-2 


54-- 


169-6 


2290-2 


60- 


188-4 


2827-4 


66— 


207-3 


3421-2 




170- 


2300-8 




188-8 


2839-2 




207-7 


3434-1 


JL _- 
4 


170-4 


2311-4 


l " 


189-2 


2851-0 


i - 


208-1 


3447-1 




170-8 


2322-1 




189-6 


2862-8 




208-5 


3460-1 


i— 


171-2 


2332-8 


1- 


190- 


2874-7 


* — 


208-9 


3473-2 




171-6 


2343-5 




190'4 


2886-6 




209-3 


3486-3 


l'r 


172- 


2354-2 


1 - 


190-8 


2898-5 


i - 


209-7 


3499-3 




172-3 


2365-0 




191-2 


2910-5 




210- 


3512-5 


55- 


172-7 


2375-8 


61- 


191*6 


2922-4 


67- 


210-4 


3525-6 




173-1 


2386-6 




192- 


2934-4 




210-8 


3538-8 


i - 


173-5 


2397-4 


i " 


192-4 


2946-4 
2958-5 


i ~ 


211-2 


3552-0 




- 173-9 


2408-3 




192-8 




211-6 


3565-2 


£- 


- 174-3 


2419-2 


*- 


193-2 


2970-5 


J— 


212- 


3578-4 




- 174-7 


2430-1 




193-6 


2982-6 




212-4 


3591-7 


1 - 


- 175-1 


2441-0 


1 - 


193-9 


2994-7 


I - 


212-8 


3605-0 




-175-5 


2452-0 




- 194-3 


3006-9 




213-2 


3618-3 


56- 


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2463-0 


62 ~~ 


- 194-7 


3019-0 


68- 


213-6 


3631-6 




- 176-3 


2474-0 




195-1 


3031-2 




214- 


3645-0 


1 - 


-176-7 


2485-0 


i- 


195-5 


3043-4 


l ~ 


214-4 3658-4 




177-1 


2496-1 




-195-9 


3055-7 




-214-8 3671-8 


*- 


-177-5 


2507-1 


*- 


196-3 


3067-9 


*- 


215-1 


3685-2 




- 177-8 


2518-2 




196-7 


3080-2 




-215-5 


3698-7 


1- 


-178-2 


2529-4 


1- 


197*1 


3092-5 


i- 


-215-9 


3712-2 




-178-6 


2540-5 




-197*5 


3104-8 




216-3 


3725-7 


57- 


-179- 


2551-7 


63- 


197-9 


3117-2 


69- 


-216-7 


3739-2 




- 179-4 


2562-9 




198-3 


3129-6 




217-1 


3752-8 


i - 


- 179-8 


2574-1 


t~ 


-198-7 


3142-0 


i " 


217-5 


3766-4 




- 180-2 


2585-4 




199- 


3154-4 




217-9 


3780-0 


*- 


-180-6 


2596-7 


*- 


- 199-4 


3166-9 


1- 


218-3 


3793-6 




-181- 


2608-0 




199-8 


3179-4 




- 218-7 


3807-3 


1 - 


- 181-4 


2619-3 


* ^ 


-200-2 


3191-9 


1 - 


-219-1 


3821-0 




-181-8 


2630-7 




-200-6 


3204-4 




-219-5 


3834-7 


58- 


-182-2 


2642-0 


64- 


■201- 


3216-9 


70- 


-219-9 


3848-4 




-182-6 


2653-4 




-201-4 


3229-5 




-220-3 


3862-2 


J - 


-182-9 


2664-9 


!■- 


-201-8 


3242-1 


i - 


-220-6 


3875-9 




- 183-3 


2676-3 




-202-2 


3254-8 




- 221- 


3889 8 


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- 183-7 


2687-8 


h- 


-202-6 


3267-4 


l- 


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3903-6 




- 184-1 


2699-3 




-203- 


3280-1 




-221-8 


3917-4 


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- 184-5 


2710-8 


1 - 


-203-4 


3292-8 


l - 


- 222-2 


3931-3 


- 184-9 


2722-4 




■203-8 


3305-5 




■222-6 j 3945-2 



108 



Circumferences and Areas op Circles. 



Circ. Area. 


Circ. Aren. 


Circ. ! Area. 


Diame- / y{ % 


Diame- ( \ \t 
to, ^J\ 


Diame- /^~"\ ' dHH 


tar. ^ 


IIP 


to, i^J 


IIP 


H-n 


223- 


3959-2 


77-i 


241-9 4656-6 


83 -r 


,260-7 


5410-6 




223-4 


3973-1 




242-2 4671-7 


- 


261-1 


5426-9 


I- 


223-8 


3987-1 


* " 


242-6 J46S6-9 


i -' 


261-5 


5443-2 




224-2 


4001-1 




243- 14702-1 


261-9 


5459-6 


*- 


224-6 


4015-1 


*- 


243-4 ! 4717*3 


*— 


262-3 


5476-0 




225- 


4029-2 




243*8 4732-5 




262-7 


5492-4 


1 -- 


225-4 


4043-2 


i- 


244-2 


4747*7 


1 ~ 


263-1 


550S-8 




225-8 


4067-3 




244-6 


4763*0 


84 — 


263*5 


5525-3 


72-- 


226-1 


4071-5 


73-— 


245- 


4778-3 


263*8 


5541-7 




226-5 


4085-6 


- 


245-4 


4793-7 




264*2 


5558-2 


i 4 


226-9 


4099-8 


i - 


245-8 


4809-0 


I - 


264-6 


5574-8 


, - 


227-3 


4114-0 




246-2 


4824-4 




265* 


5591-3 


227-7 


4128-2 


1- 


246-6 


4839-8 


I' 


265-4 


5607-9 




228-1 


4142-5 




247- 


4855-2 




265-8 


5624-5 


i - 


228-5 


4156-7 


I - 


247-4 


4870-7 


1 


266-2 


5641-1 




228-9 


4171-0 




247-7 


4SS6-1 




266-6 


5657-8 


73 — 


^ 229-3 


4185-3 


79— 


248-1 


4901-6 


85— 


267* 


5674-5 




'229-7 


4199-7 


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248-5 


4917-2 


267*4 


5691-2 


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J 230-l 


4214-1 


V; 


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4932-7 


i -- 

4 


J267-8 


5707-9 




-'230-5 


422S-5 


J 249-3 


494S-3 




268-2 


5724-6 


k- 


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4242-9 


4— 


249-7 


4963-9 


i~ 


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5741-4 




-231-3 4257-3 


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250-1 4979-5 


268-9 


5758-2 


£ - 


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1 " 


250-5 ! 4995-1 


1 - 

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269-3 


5775-0 




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250-9 5010-8 


269-7 


5791-9 


74- 


4232*4 ,4300-8 


80— 


251-3 5026-5 


270-1 


5808-8 


■ 


h232-8 4315-3 
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251-7 5042-2 




270-5 


5825-7 


i - 


l- 


252-1 5058-0 


i T 


■270-9 


5842-6 




^233-6 4344-5 




252-5 ! 5073-7 




271-3 


5859-5 


£- 


J 234* 4359-1 


i— 


-252-8 


50S9-5 


i~ 


-271-7 


5876-5 


£ - 


-j 234-4 4373-8 


- 


253-2 
253-6 


5105-4 




- 272-1 


5S93-5 


-234-8 4388-4 


i- 


5121-2 


1 " 


-272-5 


5910-5 




-1235-2 ! 4403-1 




254- 


5137-1 




-272-9 


5927-6 


75- 


^235-6 14417-8 


81— 


254-4 


5153-0 


87— 


- 273-3 


5944-6 




-236- 


4432-6 




254-8 


5168-9 




-273-7 


5961-7 


i - 


4236-4 


4447-3 


i- 


255-2 


5184-8 


1 " 


274-1 


5978-9 




-,236-7 


4462-1 




-255-6 5200-8 




- 274-4 


5996-0 


*- 


r 237*1 


4476-9 


h- 


-256- 15216-8 


i- 


- 274-8 


,6013-2 




-237-5 


4491-8 




-256-4 


5232-8 




- 275-2 


6030-4 


1 - 


.237-9 


4506-6 


£ - 


-256-S 


5248-8 


1 -1 


-275-6 


6047-6 




4238-3 


4521-5 




-257-2 


! 5264-9 




-276- 16064-8 


76- 


-238-7 


4536-4 


82- 


-257-6 


' 5281-0 


88- 


-276-4 6082-1 




-239-1 


4551-4 




-25S- 


I 5297-1 




-276-8 6099-4 


i - 


^239-5 


'4566-3 


JL ,j 

■ 4 


-258-3 


5313-2 


i " 


-277-2 6116-7 




-J 239-9 


■4581-3 




-258-7 


5329-4 




-277-6 6134-0 


*- 


-1240-3 


',4596-3 


*- 


-259-1 


5345-6 


*- 


- 278- 6151-4 




4240-7 


! 4611-3 




-259-5 


5361-8 




- 278-4 6168-8 


1 - 


241-1 


* 4626-4 


1 - 


-259-9 


5378-0 


1 ' 


- 278-8 6186-2 




-241-5 


4641-5 




-260-3 


| 5394-3 


■ J 


■ 279-2 1 6203-6 






1 _ 




J i 



Circumferences and Areas of Circles. 




Diame- ( \ 
ter. \^J 

93 -i 292-1 
i292-5 



Circ. Are?. 



i-t 

94- 

95 



96-r 



293-3 
293-r 
294-1 
294-5 
294-9 
295-3 
295-7 
296- 

■4 

296-8 

297-2 

297-6 

H298- 

298-4 
-1,298-8 

i--r 299-2 
299-6 
300- 
800-4 

| -f|300-8 
-301-2 



301-5 
301-9 
302-3 
302-7 
303-1 
303-5 
303-9 
304-3 



6792-9 
6811-1 
6829-4 
6847-8 
6866-1 
6884-5 
6902-9 
6921-3 
6939-7 
6958-2 
6976-7 
6995-2 
7013-8 
7032-3 
7050-9 
7069-5 
7088-2 
7106-9 
#125-5 
7144-3 
7163-0 
7181-8 
7200-5 
7219-4 
7238-2 
7257-1 
7275-9 
7294-9 
7313-8 
7332-8 
7351-7 
7370-7 



97- 


- 304-7 




-305-1 


l - 


-305-5 




-305-9 


H 


306-3 




J306-6 


I - 


307' 




307-4 


98— 


307*8 




308-2 


I - 


308-6 




309-0 


*~l 


309-4 




309-8 


1 - 


310-2 




310-6 


99- 


311-0 




311-4 


i 4 


311-8 




312-1 


*~ 


312-5 




312-9 


i - 


313-3 




313-7 


100— 


314-1 




314-5 


I " 


314-9 




315-3 


*-- 


315-7 




316-0 


t~ 


316-4 



103 

Circ. I Area. 



3S9-S 
I740S-8 
7427-9 
7447-0 
7466-2 
7485-3 
7504-5 
7523-7 
7542-9 
7562-2 
7581-5 
7600-8 
7620-1 
7639-4 
7658-8 
7678*2 
7697-7 
7717-1 
7736-6 
7756-1 
7775-6 
7795-2 
7814-7 
7834-3 
7853-9 
7853-6 
7893-3 
7913-1 
7932-7 
7942-4 
7972-2 
316-8 ! 7991-9 



EXPLANATION OF THE TABLE FOR SEGMENTS, &c. 

The chord divided by tlie heigkt is tlie gauge in the Table, the quotient in the 
first column. 

k = tabular coefficient, always to be multiplied by the chord. 

To find tlie angle of an arc of a circle* 

RULE. Divide the base (chord) of the arc by its height, (sine verse) and find 
the quotient in the first column. The corresponding number in the second 
column is the angle of the arc in degrees of the circle. 

To iind the radius of an arc of a circle* 

RULE. Divide the chord of the arc by its height, and find the quotient in 
the first column. The corresponding number in "the third column, multiplied 
by the chord, is the radius of the arc. 



110 



Table for Segments &c, of a Circle. 



Chord div. 


Centre 


Radius 


Cir. Arc. 


Area Seg. 


1 Surface 


Solidity 


Chord 


Ly height. 


Angle v. 


r = kc. 


£ = *c 


a =h a. 


a = A c*. 


C=» h C 3 . 


c= hr. 


<^ 


<P 


O 


^Iii^il| 


? ^^^'^^ 


\r 


\s 


458-08 


1 


57-296 


1-0000 


•01091 


•78539 


•00085 


•01744 


229-18 


2 


28-649 


1-0000 


•00218 


•78549 


•00172 


•03490 


152-77 


3 


19-101 


1-0000 


•00327 


•78462 


•00255 


•05234 


114-57 


4 


14-327 


1-0000 


•00436 


•78574 


•00310 


•06978 


84-747 


5 


11-462 


1-0001 


.00647 


•7S586 


•00401 


•0S722 


76-375 


6 


9-5530 


1-0003 


•00741 


•7S599 


•00514 


•10466 


65-943 


7 


8-1902 


1-0004 


•00910 


•78621 


•00592 


•12208 


57-273 


8 


7-1678 


1-0006 


•010S9 


•78630 


•00686 


•13950 


50-902 


9 


6-3728 


1-0008 


•01254 


•78665 


•00772 


•15690 


45-807 


10 


5-7368 


1-0011 


•01407 


•78695 


! -00857 


•17430 


41-203 


11 


5-2167 


1-0013 


•01552 


•78730 


j -00964 


•19168 


38-133 


12 


4-7S34 


1-0016 


•01695 


•78725 


-01031 


•20904 


35-221 


13 


4-4168 


1-0019 


•01S41 


•78794 


•01114 


•22640 


32-742 


14 


4-1027 


1-0023 


•02000 


•78832 


•01199 


•24372 


30-514 


15 


3-S307 


1-0027 


•02157 


•7S8S9 


•01288 


•26104 


28-601 


16 


3-5927 


1-0029 


•02269 


•78909 


•01375 


•27834 


26-915 


17 


3-3827 


1-0034 
1-0T»39 


•02434 


•78969 


•01462 


•29560 


25-412 


18 


3-1962 


•02592 


•79028 


•01542 


•31286 


24-06S 


19 


3-0293 


1-0044 


•02744 


•79084 


•01635 


•3300S 


22-860 


20 


2-S793 


1-004S 


•02878 


•79140 


•01722 


•34728 


21-760 


21 


2-7440 


1-0054 


•03040 


•79234 


•01802 


•36446 


20-777 


22 


2-6222 


1-0059 


•03178 


•79300 


•01897 


•38160 


19-862 


23 


2-5080 


1-0066 


•03343 


•79340 


•019S4 


•39S72 


19-028 


24 


2-4050 


1-0072 


•03493 


•79416 


•02072 


•41582 


18-261 


25 


2-3101 


1-0078 


•03639 


•79486 


•02159 


•43286 


17-553 


26 


2-2233 


1-00S4 


•03784 


•79530 


•02248 


•44990 


16-970 


27 


2-1418 


1-0091 


•03970 


•79639 


•02315 


•46688 


16-288 


28 


2-0673 


1-0101 


•04115 


•79748 


•02424 


•48384 


15-721 


29 


1-9969 


1-0105 


•04230 


•79811 


•02511 


•50076 


15-191 


30 


1-9319 


1-0113 


•04385 


•79907 


•02600 


•51762 


14-970 


31 


1-8710 


1-0121 


•04476 


•78530 


•02692 


•53446 


14-230 


32 


1-S140 


1-0129 


•04710 


•80098 


•02778 


•55126 


13-796 


33 


1-7605 


1-013S 


. -04842 


•80181 


•02S66 


•56802 


13-382 


34 


1-7102 


1-0146 


•04989 


•S0300 


•02956 


.58479 


12-994 


35 


1-6628 


1-0155 


•05137 


•S0405 


•03046 


•60140 


12-733 


36 


1-6184 


1-0167 


•05311 


•80531 


•03137 


•61802 


12-473 


37 


1-5758 


1-0174 


•05401 


•80622 


•03226 


•63460 


11-931 


38 


1-5358 


1-0184 


•05628 


•80713 


•03328 


•65112 


11-621 


39 


1-4979 


1-0194 


•05755 


•S0S50 


•03418 


•66760 


11-342 


40 


1-4619 


1-0204 


•05899 


•S09S7 


•03506 


•68404 


11-060 


41 


1-4268 


1-0207 


•06001 


•S1046 


•035S9 


•70040 


10-791 


42 


1-3952 


1-0226 


•06196 


•81240 


•03680 


•71672 


10-534 


43 


1-3643 


1-0237 


•06359- 


•81377 


•03773 


•73300 


10-289 


44 


1-3347 


1-024S 


•06574 


•81505 ! 


•03864 


•74920 


10-043 


45 


1-3066 


1-0260 


•06628 


•S1756 i 


•03890 


•76536 


9-8303 


46 


1-2797 : 


1-0272 


•06S26 


.81795 


•04050 


•78146 


9-6153 


47 


1-2539 i 


1-0290 


•06998 


•81939 


•04143 


•79748 


9-4092 


48 


1-2289 j 1-0297 1 


•09138 


•82064 


•04247 


•81346 







Table tox Segments &c, oi 


a Circle 




111 


Chord div. 


Centre 


Rad ius 


Cir. Arc. 


Area Seg. 


1 Surface 


Solidity 1 Chord 


by heidit. 


Angle v. 


r = k c. 


b = ft c. j 


a = h. c». 


L=£c». 


C = k c 3 . 


' c = kr. 


^'"~~~~" . 


4?0gifeh 




N/* 


XX 


S^ 


Ns/ 


N ^' 


\S 


N/ 


9-2113 


49 


1-2057 


1-0309 


•07290 


•82244 


•04330 


•82938 


9-0214 


50 


1-1831 


1-0323 


•07453 


•82384 


•04424 


•84522 


8-8337 


51 


1-1614 


1-0336 


•07611 


•S25Q2 


•04519 


•86102 


8-6629 


52 


1-1406 


1-0349 


•07758 


•82729 


•04614 


•87674 


8-4462 


53 


1-1206 


1-0364 


•07959 


•83363 


•04685 


•89238 


8-3306 


54 


1-1014 


1-0378 


•0S083 


•83072 


•04805 


•90793 


8-1733 


55 


1-0828 


1-0393 


•08246 


•83249 


•04901 


•92348 


8-0215 


56 


1-0650 


1-0407 


•08400 


•S3422 


•05002 


•93S94 


7-8750 


57 


1-0478 


1-0422 


•0S579 


•S3602 


•05098 


•95430 


7-7334 


58 


1-0313 


1-0431 


•03680 


•S3796 


•05191 


•96960 


7-5895 


59 


1-0154 


1-0454 


•03S91 


•81064 


•05299 


•98484 


7*4565 


60 


1-0000 


1-0470 


•09106 


•84266 


•05400 


1-0000 


7*3358 


61 


•98515 


1-0486 


•09209 


•84380 


•054P6 


1-0150 


7-2118 


62 


•97080 


1-0503 


•09375 


•84581 


•05583 


1-0300 


7-0914 


63 


•95694 


1-0520 


•09540 


•84791 


•05684 


1-0450 


6-9748 


64 


•94352 


1-0537 


•09697 


•84993 


•05784 


1-0598 


6-S616 


65 


•93058 


1-0555 


•09865 


•85215 


•05885 


1-0746 


6-7512 


66 


•91804 


1-0573 


•10036 


•85441 


•05987 


1-0S92 


6-6453 


67 


•90590 


1-0591 


•10201 


•85640 


•060S8 


1-1038 


6-5469 


6S 


•89415 


1-0610 


•10367 


•85S15 


•06181 


1-1184 


6-4902 


69 


•88276 


1-0629 


•10520 


•85464 


•06201 


1-1328 


6-3431 


70 


•87172 


1-0648 


•10710 


•8G350 


•06396 


1-1471 


6*2400 


71 


•86102 


1-8668 


•10SS7 


•86699 


•06515 


1-1614 


6-1553 


72 


•85065 


1-06S7 


•11046 


•86S34 


•06604 


1-1755 


6-0652 


73 


•84058 


1-0708 


•11225 


•870S1 


•06709 


1-1896 


5-9773 


74 


•83082 


1-0728 


•11385 


•87935 


•06315 


1-2036 


5-8918 


75 


•82134 


1-0749 


•11563 


•87590 


•06921 


1-2175 


5-8084 


76 


•81213 


1-0770 


•11736 


•87853 


•07037 


1-2313 


5-7271 


77 


•80319 


1-0792 


•11910 


•8S120 


•07136 


1-2450 


5-6478 


78 


•79449 


1-0814 


•12072 


•88389 


•07244 


1-2586 


5-5704 


79 


•78606 


1-0836 


•12281 


•88677 


•07352 


1-2721 


5^949 


80 


•77786 


1-0859 


•12441 


•8S949 


•07462 


1-2855 


5-4254 


81 


•76983 


1-0882 


•12660 


•S9161 


•07512 


1-2989 


5-3492 


82 


•76212 


1-0905 


•12793 


•S9520 


•07683 


1-3121 


5-2705 


83 


•75458 


1-0920 


•12958 


•89958 


•07819 


1-3252 


5-2101 


84 


•74724 


1-0953 


•13157 


•90095 


•07907 


1-3383 


5-1429 


85 


•74009 


1-0977 


•13330 


•90420 


•07960 


1-3512 


5-0772 


86 


•73314 


1-1012 


•13546 


•90734 


•08102 


1-3639 


5-0134 


87 


•72637 


1-1027 


•13704 


•91036 


•08440 


1-3767 


4-9501 


88 


•71978 


1-1054 


•13893 


•91363 


•08836 


1-3893 


4-8886 


89 


•71336 


1-1079 


•14078 


•91696 


•08450 


1-4818 


4-8216 


90 


•70710 


1-1105 


•14279 


•92210 


•08621 


1-4142 


4-7694 


91 


•70101 


1-1132 


•14449 


•92352 


•08716 


1-4265 


4-7117 


92 


•6950S 


1-1159 


•14643 


•92476 


•0S798 


1-4387 


4-6615 


93 


•68930 


1-11S6 


•14817 


•92914 i 


•08932 


1-4507 


4-5999 


94 


•68366 


1-1211 


•15009 


•933S5 i 


•09076 


1-4627 


4-5453 


95 . 


•67817 


1-1242 


•15211 


•93746 


•09197 


1-4745 


4-4845 


96 


•67282 ! 1-1271 


•15375 


•94272 ' 


•09348 1 1-4863 



112 




Table Ft 


n Sp-omenis &c., of 


a Circle. 






Chord div. 


Centre * j 


Rad ius 


Cir.Arc. 


Area Seg . 


Surface 


Solidity | Chord 


bv heisrlit. 


Angle v. 


r = Ac 


b = ke. 


a = k c'*. 


a = k c 3 . 


C = * c s . 


c = kr. 


3> 


^ 


^ 


X? 


L—^ 




r S 


4-4393 


97 


■66760 


1-1300 


•15600 


•94470 


•09442 


1-4979 


4-3359 


93 


•66250 


1-1329 


•15801 


•94852 


•09567 


1-5094 


4-3333 


99 


•65754 


1-1359 


•15995 


•95236 


•09693 


1-5208 


4-2362 


100 


65270 


1-1382 


•16180 


•95682 


•09831 


1-5321 


4-2406 


101 


64798 


1-1420 


.16393 


•96011 


•09856 


1-5432 


4-1930 


102 


•64338 


1-1451 


•16610 


•96412 


•10076 


1-5543 


4-1570 


103 


63889 


1-1483 


•16925 


•96568 


•10557 


1-5652 


4-1006 


104 


63450 


1-1515 


•17001 


•97246 


•10273 


1-5760 


4-0555 


105 


•63023 


1-1547 


•17204 


•97643 


•10471 


1-5S67 


4-0113 


106 


•62607 


1-1530 


•17414 


•98067 


•10601 


1-5973 


3-9679 


107 


62200 


1-1614 


•17619 


•9S495 


•10735 


1-6077 


3-9252 


108 


•61803 


1-1648 


•17832 


•98931 


•10870 


1-6180 


3-SS32 


109 


61416 


1-1682 


•1S041 


•99376 


•11007 


1-6282 


3-8419 


110 


61039 


1-1716 


•18257 


•93S27 


•11149 


1-6388 


3-S013 


111 


60670 


1-1752 


•18472 


1-0028 


•11234 


1-6482 


3-7612 


112 


60325 


1-1790 


•18696 


1-0077 


•11426 


1-6581 


3-7221 


113 


59960 


1-1823 


•18900 


1-0122 


•11566 


1-6677 


3-6S37 


114 


59618 


1-1859 


•19117 


1-0169 


•11709 


1-6773 


3-6454 


115 


59284 


1-1897 


•19339 


1-0218 


•11853 


1-6S67 


3-6086 


116 


58959 


1-1934 


•19559 


1-0266 ! 


•11995 


1-6961 


3-5712 


117 


5S641 


1-1972 


•19787 


1-0317 ! 


•12145 


1-7053 


3-5349 


118 


58331 


1-2011 


•20009 


1-0368 


•12294 


1-7143 


3-4992 


119 


58030 


1-2050 


•20227 


1-0417 


•12444 


1-7232 


3-4641 


120 


57735 


1-2089 


•20453 


1-0472 


•12596 


1-7320 


3-4296 


121 


'57450 


1-2130 


•20678 


1-0525 


•12748 


1-7407 


3-3953 


122 


57168 


1-2177 


•20945 


1-0578 


•12903 


1-7492 


3-3616 


123 


56895 


1-2213 


•21175 


1-0634 


•13060 


1-7576 


3-32S5 


124 


56628 


1-2253 


•21399 


1-0690 


•13218 


1-7659 


3-2940 


125 


56370 


1-2295 


•2153S 


1-0753 


•13391 


1-7740 


3-2637 


126 


56116 


1-2338 


•21859 


1-0803 


♦1355S 


1-7820 


3-2319 


127 


55870 


1-2331 


•22121 


1-0S62 


•13701 


1-7898 


3-2006 


128 


55630 


1-2425 


•22370 


1-0921 


•13866 


1-7976 


3-1716 


129 


55396 


1-2470 


•22617 


1-0974 


•14028 


1-8051 


3-1393 


130 


55169 


1-2515 


•22S65 


1-1040 


•14202 


1-8126 


3-1093 


131 


54947 


1-2561 


•23113 


1-1104 


•14371 


1-3199 


3-0805 


132 


54732 


1-2607 


•23372 


1-1164 


•14537 


1-8271 


3-0555 


133 


54522 


1-2654 


•23603 


1-1212 j 


•14676 


1-S341 


3-0216 


134 


54318 


1-2701 


•23892 


1-1295 1 


•14894 


1-8410 


2-9777 


135 


54120 


1-2749 


•24198 


1-1420 ! 


•15209 


1-8477 


2-9651 


136 


53927 


1-2793 


•24364 


1-1428 


•15252 1-8543 


2-9374 


137 


53740 


1-2847 


•24676 


1-1495 


•15422 1-8608 


2-9115 


133 


53557 


1-2897 


•24938 


1-1558 


•15605 


1-8671 


2-8829 


139 


533S0 


1-2948 


•25222 


1-1634 i 


•15807 


1-8733 


2-S562 


140 


53209 


1-2999 


•25485 


1-1705 


•15996 


1-8794 


2-8299 


141 


53042 


1-3051 


•25759 


1-1777 


•16201 


1-8853 


2-8033 


142 


52881 


1-3065 


•25936 


1-1S51 


•163S1 


1-8910 


2-7781 


143 


52724 


1-3157 


•26320 


1-1925 


•16577 


1-8966 


2-7527 


144 


52573 


1-3211 


•26604 


1-2000 


•16776 


1-9021 



Table for Seomext3 &c, of a Circle. 



113 



Choid div. 


Centre 


Radius 


Cir. Arc. 


Area Seg. 


Surface 


Solidity 


— — i 

Chord 


by height. 


Angle V. 


r = \c. 


b = k c. 


a= k a*. 


a = k C*- 


B = k c s . 


c = kr. 


^^ 


<v? 


<P 


•<:;/ 


am. 


'"/ J "^\f>\ ffftf^B^ 


v/ 


- 


X/' 


e * 


2-7276 


145 


•52426 


1-3265 


•268S9 


1-2077 


•16965 


1-9074 


2-7002 


146 


•52284 


1-3320 


•27196 


1-2166 


•17209 


1-9126 


2-6816 


147 


•52147 


1-3377 


•27449 


1-2219 


•17205 


1-9176 


2-6533 


148 


•52015 


1-3433 


•27772 


1-2318 


•17605 


1-9225 


2-6301 


149 


•51887 


1-3491 


•28168 


1-2396 


•17809 


1-9272 


2-6064 


150 


•51764 


1-3549 


•28369 


1-2476 


•18023 


1-9318 


2-5830 


151 


•51645 


1-3608 


•28674 


1-2563 


•18666 


1-9363 


2-5598 


152 


•51530 


1-3668 


•289S3 


1-2648 


•18751 


1-9406 


2-5239 


153 


•51420 


1-3729 


•29397 


1-2S01 


•18845 


1-9447 


2-5143 


154 


•51315 


1-3790 


•29607 


1-2824 


•18913 


1-9487 


2-4919 


155 


•51214 


1-3852 


•29928 


1-2914 


•19147 


1-9526 


2-4699 


156 


•51117 


1-3919 


•30259 


1-3004 


•19374 


1-9563 


2-4478 


157 


•51014 


1-3973 


•30560 


1-3094 


•19607 


1-9598 


2-4262 


158 


•50936 


1-4043 


•30905 


1-3191 


•20029 


1-9632 


2-4047 


159 


•50851 


1-4109 


•31239 


1-3287 


•20095 


1-9663 


2-3835 


160 


•50771 


1-4175 


•31575 


1-3368 


•20342 


1-9696 


2-3613 


161 


•50695 


1-4243 


•31931 


1-3490 


•20609 


1-9725 


2-3417 


162 


•50623 


1-4311 


•32263 


1-3583 


•20847 


1-9753 


2-3211 


163 


•50555 


1-4380 


•32618 


1-3682 ; 


•21105 


1-9780 


2-3004 


164 


•50491 


1-4450 


•32969 


1-3791 ; 


•21371 


1-9805 


2-2805 


165 


•50431 


1-4520 


•33327 


1-3895 


•21634 


1-9829 


2-2605 


166 


•50374 


1-4592 


•33684 


1-4021 


•21904 


1-9S51 


2-2408 


167 


•50323 1-4665 


•34048 


1-4111 


•22177 


1-9871 


2-2212 


168 


•50275 1-4739 


•34422 


1-4222 


•21946 


1-9890 


2-2013 


169 


•50231 


1-4813 


•34802 


1-4344 


•22766 


1-9908 


2-1826 


170 


•50191 


1-4889 


•35230 : 1-4476 


•23028 


1-9924 


2-1636 


171 


•50154 


1-4966 


•35563 1-4565 


•23266 


1-9938 


2-1447 


172 


•50122 


1-5044 


•35953 1-4684 


•23650 


1-9951 


2-1271 


173 


•50093 1-5123 


•36337 1-4797 


•23900 


1-9962 


2-1075 


174 


•50068 1-5202 


•36747 


1-4927 


•24225 


1-9972 


2-0892 


175 


•50047 


1-5283 


•37152 


1-5052 


•24537 


1-9981 


2-0710 


176 


•50030 


1-5365 


•37562 


1-5179 


•24856 


1-9988 


2-0530 


177 


•50017 


1-5448 


•37974 


1-5308 


•25179 


1-9993 


2-0352 


178 


•50007 


1-5533 


•38401 


1-5439 


•25531 


1-9996 


2-0175 


179 


•50002 


1-5618 


•38828 


1-5573 


•25840 1-9999 


2-0000 


I 180 


•50000 


1-5707 


•39269 


1-5708 | -26179 2-0000 




To fi 


Bid tlie length of an arc of a circle* 


RULE 


Divide t 


ie chord of the arc by its height, and find the quotient in 


the first 


column. ' 


rhe corresponding number in the fourth eolumn multiplied 


by the c 


tiord is the 
To fii 


length of the arc. 

nd the area of a segment of a circle^ 


RULE 


. Divide t 


he chord of the segment by its height, and find the quotient 


in the fi 


rst column 


. The corresponding number in the fifth column multiplied 


| "by the s< 

1 


juare of th 


e chord, is the area of the segment. 




10 • 




T 


[ 









114 



Coefficient for Capacity and Weight. 



"H 



Coefficient for Capacity and WcigSit, 



Names of Substances. 
Cubic inches, - - 
Cubic feet, - - - 
Gallons, - - - - 
Water, fresh, - - 
Water, salt, - - - 

Oil, 

Cast-iron, - - - 
Wrought-iron, - - 

Steel, 

Brass, - - - - - 

Tin, 

Lead, - - - - - 

Zinc, 

Copper, - - - - 
Mercury,. - - - 
Stone, common, - 
Clay, ..... 
Earth, compact, - 
Earth, loose, - - 
Oak, dry, - - - 

Pine, 

Mahogany, - - - 
Coal, stone, - - - 
fjliarcoal. - - - - 





FFF. 


Fit. 


1728 


12 


1 


-..694 


7-476 


0-052 


625 


0-433 


64-3 


0-445 


57-5 


0-4 


450 


3-12 


487 


3-37 


490 


3-4 


532 


3-68 


456 


3-16 


710 


4-92 


440 


3-05 


556 


3-85 


850 


5-9 


156 


1-08 


135 


0-936 


127 


0-88 


95 


0-66 


58 


0-4 


30 


0-208 


66 


0-457 


54 


0-375 


27*5 


0-19 j 



III. 

1 

• 58 

•...433 

0-036 

0-037 

0-033 

0-26 

0-281 

0-2S3 

0-307 

0-263 

0-41 

0-254 

0*321 

0-491 

0-09 

0-078 

0*0733 

0-055 

0-033 

0-017 

0-038 

0-031 

0-016 



FF*. 


Fi*. 


u'a. 


F*. 


1356 


9-42 


0-780 


903-7 


0-785 


-..549 


• 44 


0-523 


5-868 


• .408 


* 34 


3-91 


49 


0-34 


'.283 


32-7 


50-4 


0-35 


0-029 


33-6 


45-1 


0-313 


0-026 


30 


353 


2-45 


0-204 


235 


382 


2-G5 


0-221 


255 


385 


2-67 


0-222 


257 


417 


2-9 


0-241 


278 


358 


2-48 


0-207 


239 


557 


3-87 


0-322 


371 


345 


2-4 


0-2 


230 


436 


3-03 * 


0-252 


291 


666 


4-63 


0-385 


445 


122 


0-85 


0-071 


82 


106 


0-735 


0-061 


70 


99 


0-G92 


0-058 


66 


74 


0-517 


0-043 


50 


44 


0-316 


0-026 


30 


24 


0-103 


0-014 


16 


52 


0-30 


0-03 


34 


42 


0-294 


0-024 


2S-2 


21 


0-15 


0-012 


14-4 




i*. 

0-523 
.....3 

-..226 

0-019 

0-02 

0-017 

0-136 

0-147 

0-149 

0-161 

0-138 

0-215 

0-133 

0-168 

0-257 

0-047 

0-04 

0-038 

0-029 

0-017 

0-009 

0-02 

0-016 



To Find the Weight and Capacity toy this Table 

RULE. The product of the dimensions in feet or in inches, as noted in the 
columns, multiplied by the tabular coefficient, is the capacity of the solid, or 
weight in pounds avoirdupois. 

Example 1. A cistern is 6 feet long, 27 inches wide, and 20 inches deep. 
How many gallons of liquid can be contained in it ? 

6X27X20X0*052 = 168 48 gallons. 

Example 2. A cast-iron cylinder is 4*5 feet long, and 7"5 inches diameter- 
Required the weight of it ? 

4-54-7-5^X2-45 = 620 pounds. 





Table of 8th. Ordiisates, for Railroad Curves* 


115 


AnoJe. Ordinates. 


Angle. ! Ordinates. 


W 


1. 7. 


a. 6. 

: 00164 


3. 5. 

•00193 


•00215 


W | 1. 7. 
5 3°, *05313 


3. 6. 1 3. 5. ; 4. h. 

•08932 | -11063 j -11773 


T° 


•00084 


2 


•00191 


•00327 


•00409 


•00436 


5 4 1*05422 


•09130! -11318 j -12003 


3 


•00299 


•00522 


•00561 


•00659 


55 


•05531 


•09303 -11510 -12235 


4 


•00382 


•00654 


•00818 


•00S72 


56 


•05646 


•09487 


•11731 1-12466 


5 


•00-137 


•oosis; 


•01023 


•01091 


57 


•05760 


•09673 


•11950-12698 


6 


•00573 


•00928; 


•01228 


•01309 


58 


•05875 


•09353 


•12170; -12932 


7 


•00675 


•01173 


•01432 


•01527 


59 


•05989 


•10037 


•12393 I -13162 


8 


•00764; 


•01309, 


•01639 


•01745 


60 


•06094 


•10220 -12612 1-13397 


9 


•00845; 


•01474; 


•01842 


•01961 


61 


•06261 


•10427 


•12840 1-13631 


10 


•00955 


•01637 


•02047 


•02183 


62 


•06331 


•10593 


•13054 


•13866 


11 


•01053 


•01801 ' 


•02250 


•02402 


63 


•06451 


•10781 


•13281 


•14101 


12 


•01146 


•01965^ 


•02456 


•02620 


6 4 


•06570! 


•10964 


•13505 


•14337 


13 


•01245 


•02129 


-02662 


•02S39 


65 


•066S1 


•11101 


•13765 


•14573 


14 


•01284 


•02271 


•02361 


•03055 


6Q 


•06305 


•11342 


•13956 


•14810 


15 


•0143S 


•02461 


•03081 


•03282 


67 


•06914! 


•11532 


•14181 


•15048 


16 


•01535 


•02625 


•03277 


•0349P 


68 


•07040 j 


•11721 


•14409 


•15286 


17 


•01630 


•02789 


•034S4 


•03715 


69 


•07168 


•11912 


•14637 


•15526 


18 


•01730 ! 


•02956 


•03693 


•03935 


70 


•07234 


•12103 


•14864 


•15765 


19 


•01S5S' 


•03125 


•03996 


•04151 


71 


•07407 


•12294 


•15087 


•16005 


20 


•01922 


•03286 


•04103 


•04371 


72 


•07535 


•12485 


•15323 


•16245 


21 


•02Q22 


•03453 


•04309 


•04591 


73 


•07656 


•126S5 


'15555 


•16487 


22 


•02119 


•03619 


•04522 


•04811 


74 


•07734: 


•12877 


•15785 


•16729 


23 


•02215 


•03787; 


•04720 


•05031 


75 


•07912 


•13078 


•16016 


•16972 


1 24 


•023111 


•03934 


•04930 


•05255 


76 


•08040 


•13292 


•16247 


•17216 


25 


•02413 J 


•04117; 


•05133 


•05475 


77 


•0S16S 


•13472 


•16482 


•17460 


26 


•02508| 


•04283: 


•05346 


•05696 


78 


•08297 


•13670 


•16716 


•17706 


2 7 


•02610 ' 


•04457' 


•05552 


•05917 


79 


•0S426 


•13868 


•16951 


•17951 


28 


•02708! 


•04021 


•05761 


•06139 


80 


•0^560; 


•14070 


•17187 


•18198 


29 


•02813; 


•04793! 


•05970 


•06361 


81 


•0S695 


•14274 


•17423 


•18445 


30 


•02911! 


•04970, 


•06183 


•06582 


82 


•08829 


•14477 


•17660 


•1S694 


31 


•03005 


•05125' 


•06386 


•06801 


83 


•08944 


•14681 


•17901 


•18943 


32 


•03107: 


•0529S! 


•06596 


•07027 


84 


•09105 


•14888 


•18140 


•19193 


33 


•03191 


•05464 


•06S06 


•07250 


85 


•09235 


•15120 -1S379 


•19444 


34 


•03310 


•05637: 


•07016 


•07477 


86 


•09377 


•15304-18622 


•19695 


35 


•03412 


•05S04 


•07424 


•07695 


87 


•09518 


•1»509; -18865 


•19946 


36 


•03515 


•05992 


•07452 


•07919 


88 


•09660 


•15756 -19108 


•20201 


37 


•03616 


•06147 ! 


•07646 


•08143 


89 


•09780 


•15931! -19350 


•20555 


38 


•03718 


•06327 


•07858 


•03367 


90 


•09944 


•16144-19597 


•20710 


39 


03821 


•06492 


•0S069 


•08591 


91 


•10098 


•16359 -19842 


•20966 


40 


•03905 


•06631 


•08243 


•08816 


92 


•10240 


•16575 -20092 


•21223 


41 


•04030 


•06336 


•03494 


•09041 


93 


•10334 


•167S7 1 -20338 


•21481 


42 


•04133 


•07012 


•08707 


•09266 


94 


•10537 


•17005 1 -20589 


•21740 


43 


•04241: 


•07182 


•08920 


•09492 


95 


•10692 


•17224! -20837 


•22000 


44 


•04363! 


•07353 


•09130 


•09719 


96 


•10851 


•17444! -21091 


•22262 


45 


•01522 i 


•07531 


•09346 


•09945 


97 


•10997 


•17666-21342 


•22523 


46 


•04556' 


•07706: 


•09562 


•10172 


98 


•11150 


•178SS 1 -21596 


•22786 


47 


•01682 


■07894 


•09790 


•10400 


99 


•11310 


•18111 j -22800 


•23050 


48 


•04833] 


•0S059 


•09991 


•10627 


100 


•11468 I 


•1S354 -22107 


•23315 


49 


•04879 


•08236 


•00207 


•10856 


101 


•11626! 


•18500; -22364 


•23596 


50 


•04982 


•03413 


•00422 


•11035 


102 


•11791! 


•18793 j -22623 


•23848 


51 


•05096 


•08593 


•10639 


•11314 


103 


•11959, 


•19021-22376 


•24107 


52 


•05204 1 


•08768 


•10855 


•11543 


104 


•12116: 

I 


•19256-23147 


•24386 



116 Rail Road Curves 



BAIL 110 AD CURVES. 

When Railroads are to be connected by curves, we commonly have given the 
distance (chord c,) between the two ends oo of the tracks, and the tangential 
angle v. Ey these the curve is to be constructed. 
Example 1. Fig. 128. The chord C = 168 feet, and the tangential angle 
= 19° 30'. Required the centre angle w =, and the radius R = ? 

w = 2(19° 30') = 39°. -K = 39 fc c = 1*4979X168 = 251*647 feet. 
Jc = See Table for Segments, &c, of a circle. 

By Tangential Angles* 
The curve to be laid out by the three tangential angles r or, ron, and noo, 
each angle = ?v = 6° 30'. Required the chord r = ? 
The centre angle for the chord r is 

2X(6° 300 = 13°, and r = «& 22 = 0-2264X251-647 = 56-974 feet. 
By Angles of Deflexion* 
Divide the centre angle w into an even number of parts = z. Set off at o the 
angle z = r o n, and bisect it into ror and ron, — find the chord r, and sub-chord 
a, and continue as shown by Figure. 

Example 2. Fig. 128. The tangential angle v = 78°, and the chord C= 638 
feet. Required the centre-angie w = ? Radius R = ? Chord r = ? and the sub- 
chord a = ? 

w = 2X78° = 156°. JB = i sefc c = 0-51117X638 = 326*126 feet. 
Let the curve be laid out by 6 angles of deflexion, and z =■ ^X156° = 26°, and 
r = «fc B = 0-4499X326-126 = 146-73 feet. ' 
a = ^k r = 0-4495x1 1=6-73 == 66-012 feet. 
By Orcliiiates. 
Example 3. Fig. 129. The chord C= 368 feet, and v = 36°. Required the 
height h = ? 

h = ^C(cosecy — cot/y). 

From cosec.36° = 1-70130 

Subtract ,- - cot.36° = 1-37638 

The height h = 0-32492X184 = 59*785 feet. 

At x = 92 feet from 7i. Required the ordinate y ? 
X . 2X92 sin.36° 



- = 0-2938926 = sin.17 6'. 



368 
y = 1X36SI - 03 ' 17 °" — cot.36° )= 45-9448 feet. 



c /cos.T7° 6' 



By Siib^Chords* 

Example 4. Fig. J30. The ends o and o of the tracks form different angles w 
and TTto the chord C. and therefore must be connected by two curves of differ- 
ent radii, R a d r. The chord C= 869 feet, w = 38°, and 1F= 86°. Required 
the distance from o to the height h, n — 1 sub-chord b — ? sub-chord a = ? 
radii i2 and r = ? 

v = ix38° = 19°, and F= £XS6° = 43°. 

869tan.l9° on , OK-fi . 

w = =. 2C4-3D feet. 

tan.l9°+tan.43° 

6 = 231-35 ee(\43° = 320-42 feet. I R = 33 /t<2 = 1-5358X671-21 = 1030-2 ft. 

a = sec.l9°(869 — 231-35) = 671-21 ft. | r = sefc & = 0-73314X320-42 = 234-91 ft. 
By Eiglit Ordinates. 
Exanple 5. Fig. 133. Required 8 ordinates for a curve of chord C= 710 feet 
and the centre angle to — 69°? (See Table on the preceding page.) 
1st and 7th Ordinates 0-07168XT10 = 50-8928 feet. 
2nd " 6th " 0-11912X710 = 84-5752 " 

3rd " 5th " 0-14637X710 =-. 103-9227 " 

4th or height h 0-15526X710 = 110-2346 " 



Railroad Curves. 



717 



jjjik^ 


128. 

By angles of deflexion. 

w = 2v, R = w k C = JC cosec.v 






r = z Jc R, a = *k r = 2r sin.Jz. 




129. 

5?/ Ordinates. 
h= iC(cosec.v — cot.i?). 

y„l C (2?5£_COt.l,), 

\sm.t' / 

2a? sin.v 



sm.^; 




130. Zty Sub-chords, 
Ctdm.v 



71 == 



tan.i> rtan.7 ? 
b = nsec.7, 
o = sec.u(C — //), 



7i = 77 tan. 7, 

w? = 2i? 
17=27' 




131. 

i Parallel tracks by a reverse curve. 

Formulas same as above. 
The length o o = 2c, length of 
a circle arc / = 0-035u R. 




132. 

The greatest radius in a reverse 

curve. 
w = i(7+3y), 17 = w+7 — v, 
a = »kR, b = w kR, 



J£ = C sec.w(sln.V — ysin. a F— cos. 2 w). 




133. 



Curve by 8 Ordinates. 



The ordinates are calculated in the 
accompanying Table, the chord C = 1 or 
the unit. 

If the angle ?o is large, or there he some 
obstacle on the chord C, find the height 7i 
and lay out the curve by two or more sets 
of 8 ordinates. 



113 By Ordinates and Stjuchords. 



By Ordinates and Subchords. 

Example 6. Fig. 134. The tangents t being prolonged to where they 
meet at a, divide that angle into two equal parts, say PF=7o°. Required 
the tangents t=1 external secant S=1 chords C=1 and the angle w=1 
Radius of the curve £ = 1500 feet. 

t=R cot.75°=1500X0-26794=401-91 feet. 
Centre angle w=90— 75°=15° for half the curve. 
S=R (sec.lo-— 1) -1500 (1-0352-1) =52-8 feet. 
The chords C=k E=0'26104X 1500=391 -56 feet. 
Measure off from a the tangents and the external secant. 
Draw the chords C C, and divide them each into eight equal parts. 
In the table of ordinates under w=15° will be found the 

1st. 7th. 0-01433 K391-56-5-631, 3rd. 5th. 0-03081X391'56=12'063, 

2nd. 6th. 0-02461 X391'56=9-636, 4th. 0-03282X39-56=12-851, 

Thus by only four multiplications, 16 ordinates in the curve is obtained. 

Should there be any obstacles for the chords C. C. as is often the case 

in excavations and on embankments, a line can be drawn further in on 

the track parallel to the chord and the ordinates obtained by subtraction, 

readily understood by the Engineer. 

Ellipse by Ordinates. 

By this arrangement ellipses can be constructed of any proportions. 
One of the two axes is divided into 16 equal parts. The ordinates 
drawn and calculated as shown by the figure 135. 

Parallel Tracks by a Semi-Ellipse, 

Example 7. Fig. 136. The instrument placed at b and b', divide the 
angles W and w each into two equal parts, prolong the chords which 
will meet at «, a point in the curve. Divide the chords each into eight 
equal parts, and draw the ordinates parallel to the tracks as shown in the 
figure. The grand chord C is the unit for calculating the ordinates, 
which latter are alike on both the chords c, c'\ 

1st 2nd. 3rd. 4th. 6th. 6th. 7th. 

0-1795C 0-2058C 0-2029C 0-1S30C 0-1477C 0-1091 C 0-0586C. 

Suppose the grand chord to be C=2050 feet. 

Required the length of the 6th ordinate ? 0-1091X2050=223-655 feet. 

Tracks not Parallel by Elliptic, arc, 

Example 8. Fig. 137. Divide the angles W and w each into two equal 
parts, prolong the subchords until they intersect one another at a, which 
is a point in the curve. Divide the chord C into eight equal parts, join 
a with the 4th division and draw the other ordinates parallel thereto. 
Suppose the angles are W=18° and w = 12 J , the centre angle will be 30° 
for which the ordinates are to be calculated from the table. The chord 
C=125 feet. Required the 3rd and 5th ordinates 1 0-06188X125=7-335 feet. 
Springing of Rails. 
Example 9. Fig. 138. A rail of L=21 feet is to be curved to a radius 
of .R=1250 feet. Required the spring S=1 in sixteenths of an inch. 
24X21 2 
S = -:~" ~ = 8 ' 47 sixteenths - 

Super Elevation of the External Rail. 

Example 10. Fig. 139. A train running iVI=30 miles per hour on a 
curve of R =1550 feet radii, the gauge of the track is G =5 feet. Required 
the angle of inclination v=] and the super elevation of the external 
rail h=1 

30 2 

tan.v = = 0-0237=tan. 1° 21'. 

15X1550 

h=G sin.l° 21'=5X0'02356 =0-1178 feet, or nearly H inches. 

It is practically impossible to lay the super elevation to suit the dif- 
ferent speeds of trains. If a mean speed is taken, the faster passenger 
trains will wear the outer rail, and the slow or freight train will wear 
the inner rail. 



Raii.iioad Cuuves. 



119 





134. 

By ordinates and sub chords. 

t = R cot. W=R tan.w, W=$0—w, 

S= R (sec.w — 1) = R Ccosec. W — 1) 

C=k R. For k, see table of segments. 



135. 



Ellipse by ordinates. 

1 = 0-4840(7 5=0-9204(7 

2=06616(7 6=0-9682(7 

3=0-7808(7 7=0-9922(7 

4=0-8660(7 8 = (7 the unit. 



136. 




Parallel tracks by elliptic curve. 

h=l'ti. w=2v. 17=2 7, 
, C sin. W „ Csiii.w 

C =-0-: » C = o ^~T>- , 

2 sm.u 2 sm. 7 

See example for ordinates. 



137. 

Tracks not parallel by elliptic arc. 

Angle of the arc = W-\- w. 

Ordinates to be calculated from the table. 



138. 



Spring of Rails. 
1-5 U 



R 



-= spring m inches. 



2 4 L' 
S= — .5 — =16ths of an inch. 




139. 



Inclination of tracks in curves. 

tan.y= ^^5* h— G sin. v. 
15 it, 

Meaning of letters, see example. 



120 Excavation and Embankment. 



EXCAVATION AND EMBANKMENT. 

Example 1. The Road-way of an excavated channel is r = 15 feet, the depth 
D = 9 feet, and the breadth at the top b ~ 46£ feet. Require the slope & = ? 

Formula 6. S = - *~" ° = 1'75 or U to 1- 

Example 2. The Road way is to be r = 15, D = 18, and the slope S = 1£, 
Require the breadth b = ? and the cross-section A = ? 

Formula 4. fc = 2 X 18 X 1*25 + 15 = 60 feet. 

18 / \ 

Formula 7. A = — I 60 + 15 )= 675 square feet. 

Example 3. The Road-way is to be r = 16 feet, the slope 5= 1£, and the depth 
-D = 11 feet. Required the area of Cross-section A = ? 

Formula 9. A = 11 (11 X If + r) = 3575 square feet. 

Example 4. The Road-way r = 18 feet, slope S= li-, d = 14 feet 6 inches, and 
the length from o is 2 = 55 feet. Required the cubic contents c = ? 

Formula 11. c = 55 X M-tf 1 * 5 * 125 + ^)= 11995-676 cubic feet, divided 
by 27 = 444.28 cubic yards. 

Example 5. The Road-way is r = 16 feet, slope S = 1| feet, 2? = 17*5, tf = 7*4 
and the length L = 100 feet. Required the cubic content C = 1 

Fornul* 12. C = loo[«(H^±I±±HlXIl)+|(17-6+7-4)] 

= 44445 cubic feet, or 1645*4 cubic yards. 
The computation is executed thus . 

17-5 17-5 

7-4 • 7.4 



700 24-9 

1225 8 



129-50 199-2 

17-5* = 306-25 ) From table 
7-4*= 54-76 J of Squares. 



3) 490-51 (163:51 slope, add J 

199-2 
X 100 = 44445. cubic feet. 



Excavation and Embankment. 



121 



134 




Letters in the Formulas correspond with the Figure. 



& = cot V, 

a = D cot. v, - 
b = 2 D S + r, 



s " T5> 



* A=|-(5 + r), - 7. 

a'-5-(* + r), - 8. 

A = D{D S + r), 9. 

a. = d(dS + r), - 10. 

C = ^— +2)' 1L 

+ 5 (p + </)]. - 12. 



Letters Denote, 
A and a ^= Cross-Sections in square feet, of the excavated channel or 
embankment. 
D and d =-= depth in feet, of the Sections. 
r = width in feet of the Road- Way. 

= Base in feet of the embankment, or top breadth of the channel. 
L — length in feet, between the two Sections A and a. 

1 = length in feet, from the Section a to the point o where the ground is 
level with the road. 

C = cubic contents in feet, between A and a. 

c = cubic contents in feet, between a and o. 

8 = slope of the sides. The slope is commonly given in proportions, thus : 
" Slope = 1 i to 1," which means, that the side slopes 1| feet horizontally for 1 
fcot vertical. 

» — angle of the slope. 

11 



122 Railroads. 



TBACTION ON BO ADS. 

Letters denote. 
F= tractive force in pound avoir., necessary to overcome the rolling 

friction, and ascending inclined plains. 
M = miles peP hour of the train or force F. 

T = weight of the load in tons, including the weight of the carriages. 
On rail-roads T includes the weight of the locomotive and tender. 
t = weight of the locomotive resting on the driving wheels in tons. 
h = vertical rise in feet per 100 of inclined roads. 
b = base in feet per 100 of the inclined road or plain. 
k = tractive coefficient in pound per ton of the load T, as noted in the 

accompanying Table, under the different conditions of the road. 
A = area of one of the two cylinder pistons in a locomotive, in sq. in. 
P = mean pressure of steam in lbs. per sq. in. on cylinder pistons. 
S= stroke of pistons in feet. 
D = diameter of driving wheel in feet. 

H= actual horse power of a locomotive or the power necessary for the 

load. About 25 per cent, is allowed for friction and working pumps. 

/= adherence coefficient of the driving wheels to the rails, in pounds 

per ton of the weight t. 
n = revolutions per minute of driving wheels. 
d = continued working hours of a horse. 

v = velocity in feet per second. V = weight of a horse in pounds. 
Example 11. Fig. 140. The area of one of the two cylinder pistons in 
a locomotive is A =314 square inches, stroke of piston P=2 feet, mean- 
pressure P=80 lbs. per square inch. Driving wheels D=4 feet diameter. 
Required the tractive force F=1 of a locomotive. 

ir = ?Ji><2X80 = 1256Q lbg> the answer< 
4 
The adhesive force of the driving wheels to the rails,/;, must always 
be greater than the retractive force of the locomotive, otherwise the 
wheels will slip on the track. 

Example 12. Fig. 141. A locomotive of t=15 tons on an inclined plain 
rising h=10 feet, and the base 5=99-5 feet per 100. /=560, other dimen- 
sions being the same as in the preceding example. Required the tractive, 
retractive and adhesive forces 1 

Tractive, F= 314 X 2 X gg _ 22-4X15Xltfr=S200 lbs. 

4 

Retractive, F = 22-4X15X10=3360 lbs. 

Adhesive, F^ 56 ^ 1 ^"' 5 ^ 8358 lbs. 

100 

Consequently the locomotive can ascend the inclined plain with a 
tractive force of 8358 -3360=4998 lbs., without slip in the driving wheels. 

Example 13. Fig. 142. A train of T=200 tons is to be drawn M=20 
miles per hour on a horizontal track in good condition, fc=4. Required 
retractive force F=] 

jF = 200 (4+/20) = 1694-4 lbs. the answer. 

Example 14. Fig. 143. A train of T=150 tons is to be drawn up an in- 
clined plain of h=9 feet in 100, with a speed of M=16 miles per hour, 
&=4. Required the necessary horse power of the locomotive H=1 

H=- G ^l°° (22-4X9-T-4+/16) = 1342-144 horses. 

Example 15. Fig. 144. Required the tractive ability F=? of a horse, 
running i¥=7 miles per hour, in d=4 continued hours. 

375 

F = — = 26-8 lbs. the answer. 
7|/4 



Railways and Common Roads. 



123 




140. 



*— D * n ~~ D 



17== 



ASPn ASPM 



11000 376 D 

Adhesive force =ft. 




141 

_ AS P rto . . „ Dn 

Adhesive, — ->22"4*A. retractive. 



142 



.F= T 7 (k+ y/M). </*= Adhesive. 
71/^ T 



143. 




./** 



i^=rC22'4A+^+ v ^> <^=- 
^=f^22-4M-H-v/S> 





144. 



r=l-466 1f. 



j? 55 ° 375 i-r* * x. 

i* = T-r =:-—— ■— • ability of a horse. 



145. 



2?== r C22.4 A+^+v^. 



550 t'h 

''v<yd loo" 



- J/=0-6821 v. 



124 



Railroads. 



Example 16. Fig. 145. Required the tractive force F=1 of a load T=5-25 
tons, to be drawn M=2 miles per hour up a turnpike of h=d feet in 100, 
the road being newly laid with coarse gravel £=50. 

jF= 5-25 (22'4X9X5°X/^ = 1328-25 lbs. 
Suppose a horse to weigh t' = 1000 lbs., working continually in d =1 
hour up the turnpike. Required the tractive ability F=1 per horse. 



375 

_ 27l" 



1000X9 
lOO" 



= 97-5 lbs. 



1328*25 
Number of horses = — — — - = 13*6 say 14 horses which will be necessary 

for the load under the mentioned circumstances. In these examples it 
is necessary to take ikf>l. and d>l. 

Traction Coefficient at very slow Speed. 

On railroads in good condition, carriage axels well lubricated. 
On railroads under ordinary, not very good condition. 

On very smooth stone pavement, 12 

On ordinary street pavements in good condition. 20 

On street pavements and turnpikes. 30 

On turnpikes newlaid with coarse gravel and broken stones, 50 

On common roads in bad condition. 150 

On natural loose ground or sand. 560 

Adherence Coefficienti 

On rails of maximum dryness. 

a u very dry. 

" " under ordinary circumstances. 

" " in wet weather. 

" " with snow or frost. 
In railway curves, the retractive force is augmented so many per cent., 
as the whole train occupies degrees in the curve. 



/ 
672 

560 
450 
315 

224 



Plate I 




PlateH, 



„ 




M -> fe 




v l ■* *, -c : 

'S * «i «*.' 1 1 




$ a sSSs *! 


"3 * 0) • 














3 
« 






^ 




5 


^fe^ 












4 ^^^ 






°- 




t ^*< 


r^ 


It*. 
4; %^ 






























^ 


j^ 










s- 


%>- 




















{■ ^ 


^St| 


^ 














1" i5 


^05 


|f 




■v . r; s*a 


II" 


13^ 


\ 1 - 




^ ;...£ eg 


•5 f 38 


•M-£- 


— 




.- 


O 


# 








^s 


>^"' 


^ 




















">^ 


*f 




45 
















3? 






























^^ 


j^ 


>^ 
























> 





^ 





























Trigonometry. 125 



To Reduce Indies and Fractions thereof to Decimals of a 
Foots and vice versa* 

PLATE I. 

This is a common decimal scale on which the 12 inches of a foot are laid 
out. Any length of a foot expressed by inches and the common fractions, 
intersects its own value in decimals, and are read off as on a common diagonal 
scale with 10 to the base. 

Example 1. How much is 8| inches in decimals of a foot ? Find 8y inches 
on the rule, which will be found to intersect 6875 on the decimal scale; 
cr 8£ in. = 6-6875 feet. 

The first figure 6 is marked at the top and bottom of the scale, the second 
figure 8 is the eighth vertical (or nearly so) line from 6; the third figure 7 is the 
seventh horizontal line marked at the ends of the scale, and the fourth figure 5 
is the horizontal dotted line between 7 and 8. 

For sixteenths see the fine single rule. 

Example 2. How much is 0*526 feet in inches 1 6 % the answer. 

To Reduce Vulgar Fractions into Decimals and vice versa* 

PLATE IL 

This is a similar arrangement to the preceding one. The vulgar fractions 
are laid out so that the denominators are marked at the ends of the scale 
and the nominators on the line that joins the given denominators; by 
this arrangement the nominator of the vulgar fraction intersects its own value 
in decimals on the scale. To facilitate the operations it is best to imagine in 
which quarter the vulgar fraction is ; as *K fi is in the first, f in the second ; f in 
the third, and \% in the fourth quarter, &c, &c. ; each rule occupies a quarter 
of the scale, on which the vulgar fraction is to be found accordingly. 

Example 1. How much is ^f G in decimals? 

On the third rule it will be found at 5625 of decimals, or % 6 = 0-5625. The 
first and second figures 56 are marked as described for Plate I, and the third, 
fourth, &c, &c, figures are written down on the horizontal line of the given 
fraction. 

Example 2. What nearest vulgar fraction answers to the decimals 0*39583 ? 
^j the answer. 

These two diagrams are exceedingly useful in practice. By Plate II, Vulgar 
Fractions can be added and subtracted. 



TRIGONOMETRY. 

Trigonometry is that part of Geometry which treats of Triangles. It is di- 
vided into two parts, viz. : plane and spherical. 

Plane Trigonometry treats of triangles which are drawn (or imagined to.be) on 
a plane. Spherical Trigonometry treats of the triangles which are drawn (or im- 
agined to be) on a sphere. 

A triangle contains seven quantities, namely, three sides, three angles, and 
the surface ; when any three of these quantities are given, the four remaining 
ones can by them be ascertained, (one side or the area must be one of the given 
quantities) and the operation is called solving tJie triangle, which is only an ap- 
plication of arithmetic on Geometrical objects. 

For the foundation of the above mentioned solution, there are assumed eight 
help quantities which are called Trigonometrical functions, and are here denoted 
with their names and number, corresponding with Figure 1 ? In the accompa- 
nying Tables, the functions are calculated at every 10 minutes per degree in the 
quadrant of the circle represented by Fig. 1 ? The angle for which the functions 
are mentioned, is the opening between the two lines 7 and 2, 3, this angle is de- 
noted by the letter C, and the expression sin.C. means the line 1 compared with 
the radius rasa unit. 

ii*~~ ' 



126 Plane Trigonometry. 



Example 2. Fig. 136. An inclined plane a = 150 feet long, and c = 27 feet, the 
height over its base. What is the angle of inclination C = ? 

c 27 
Formula 14. sin.C= — — — « 0-18000. 

a 150 

Find 0-18000 in the table of sines* which will be found at 10° 30^ which is tho 
angle C nearly. 

Example 3. Fig. 137 An oblique angled triangle has the sides c = 27-6 feet, the 
angle C = 34° 10', and the angle A = 47° 40 / . How long is the side a = ? 

BbrmOa-L a- ^ , £$fl*ffy '_ 36-33 feet, the answer. 
5m. C sift .34° 10' ' 

By Logarithms. 

log. a = log.c-\-log.sin.A — logMn.C. 

c + log.27-6 = 1-.44090 

4 4- log.sin.47°40' = 1:86878 

1-30968 

O — log.sin.34° 10" ^ = 1:74942 

log. 36-4 = 1-56026, or a = 36*4 feet. 

Example. 1. Two ships of war notice a strong firing from a castle ; in order to 
be safe, they keep themselves at a distance beyond the reach of the balls from 
the castle. To measure the distance from the castle, they place the vessels 800 
yards from each other, and observe the angles between the castle and the ves- 
sels to be A = 63° 45', B = 75° 50'. What will be the two distances from the 
castle ? 

C = 180 — 63° 45' — 75° W = 40° 25'. 
To A the distance will be, 

sin.G sin. 40° 25' 

To B the distance will be, 

csin. A _ SOOXsin. 63° 45' 



sin.C sin. 40° 25' 



= 1106-6 yards. 



Example 2. From a window in the lower floor of a house which lays level with 
the foot of a tower, is observed an angle = 40° to the top of the tower. From 
another window in the upper story, in the same perpendicular as the lower 
window, the altitude of the tower is observed to be = 37° 30', which is 18 feet 
above the lower window. 

Then we have A = 90 — 40° = 50°. C = 90+87° W = 127° SO 7 . 

B = 180 — 50 — 127° 39' = 2° 30'. I = 18 feet. 
What will be the height of the tower and the distance from it? 
The distance from the lower window to the top of the tower = c. 

sin.B sin.2° SO 7 

The height of the tower = h. 

h — c. sin.A = 327*3Xsin.40° = 210 feet. 
The distance to the tower = d. 

d = c. cos.A = 327-3Xcos.40° = 250-8 feet. 



Trigonometry. 




1 Sinus abbreviated sin.C. 

2 Cosinus " cos.C. 
8 Sinus-verms ** sinv.C. 

4 Cosinus-versus ft cow. (7. 

5 Tangent " fcm.C. 

6 Cotangent " co^.Cl 

7 Secant " sec.C. 

8 Cosecant " cosec.C. 

r == Radius of the circle, which is the unit by which the functions are mea- 
sured. 



tan.C 



tan. C = 



cot.C = 



cot.C 



r 2 = sin.*C+cos. 2 C. 
sin.C 



cos.C' 

1- 

cot.C 

cos. C. 
sin.C 

1 

tan.C 



1 

c^sTC" 5 



sec. C 

cosec.C=- . * - , 

sm.C ' 

sinv.C =1 — cos.C, 

cosv. C = 1 sin. C, 

sin.2C = 2 sin.C cos.C, 

sin. JC = i>^sin. 2 C+sinv. 2 C), 

sin.(C+£) = sin.C cos. B± 
sin.2?cos. C. 

Positive and Negative Signs. 



Angles. 
4-0° 

-r90° 

+18G° 
'+27GO 
+360° 



sin. 


COS. 


sinv. 


CO V. 


tan. 


cot. 


sec. 


+0 


+1 


+0 


+1 


+0 


+ 00 


+1 


+1 


+0 


+1 


+0 


+00 


+0 


+00 


±0 


-1 


+2 


+1 


+o 


+oo 


—1 


—1 


+° 


+1 


+2 


±00 


+'- 


+00 


+° 


+1 


4-0 


+1 


+° 


—CO 


. +1 



cosec. 

+00 

+ 1 

+00 

—1 

—oo 



L 



"When a quantity has reached or 00, it has ceased to exist, because it can 
not be increased or diminished. 
Example. "What is the length of the secant for an angle of 74° 18'? 

Secant C = C0Si ^o^ ^S-695. 



128 








Tfjgonometrt. 








Natural Sine* 


Beg. 


0' 


j 10 ' 


20' 


30' 


40' 


50' 


60' 







•ooooo 


1 -00291 


•00581 


•00872 


•01163 


•01454 


•01745 


89 


1 


•01745 


| -02036 


•02326 


•02617 


•02908 


•03199 


•03489 


88 


2 


•034S9 


•03780 


•04071 


•04361 


•04652 


•04943 


•05233 


87 


3 


•05233 


•05524 


•05814 


•06104 


•06395 


•06685 


•06975 


86 


4 


•06975 


•07265 


•07555 


•07845 


•08135 


•08425 


•08715 


85 


5 


•08715 


•09005 


•09294 


•09584 


•09874 


•10163 


•10452 


84 


6 


•10452 


•10742 


•11031 


•11320 


•11609 


•11898 


•12186 


83 


7 


•121S6 


•12475 


•12764 


•13052 


•13340 


•13629 


•13917 


82 


8 


•13917 


•14205 


•14493 


•14780 


•15068 


1 -15356 


•15643 


81 


9 


•15643 


•15930 


•16217 


•16504 


•16791 


1 -17078 


•17364 


80 


10 


•17364 


•17651 


•17937 


•18223 


•18509 


•18795 


•19080 


79 


11 


•19080 


•19366 


•19651 


•19936 


•20221 


•20506 


•20791 


78 


12 


•20791 


•21075 


•21359 


•21643 


•21927 


•22211 


•22495 


77 


13 


•22495 


•22778 


•23061 


•23344 


•23627 


•23909 


•24192 


76 


14 


•24192 


•24474 


•24756 


•25038 


. -25319 


•25600 


•25881' 


75 


15 


•25S81 


•26162 


•26443 


•26723 


•27004 


•27284 


•27563 


74 


16 


•27563 


•27843 


•2S122 


•28401 


•28680 


•2895S 


•29237 


73 


17 


•29237 


•29515 


•29793 


•30070 


•30347 


•30624 


•30901 


72 


18 


•30901 


•31178 


•31454 


•31730 


•32006 


•32281 


•32556 


71 


19 


•32556 


•32831 


•33106 


•33380 


•33654 


•33928 


•34202 


70 


20 


•34202 


•34475 


•34748 


•35020 


•35293 


•35565 


•35836 


69 


21 


•35S36 


•36108 


•36379 


•36650 


•36920 


•37190 


•37460 


68 


22 


•37460 


•37730 


•37999 


•38263 


•38536 


•38805 


•39073 


67 


23 


•39073 


•39340 


•39607 


•39874 


•40141 


•40407 


•40673 


66 


24 


•40673 


•40939 


•41204 


•41469 


•41733 


•41998 


•42261 


65 


25 


•42261 


•42525 


•427S8 


•43051 


•43313 


•43575 


•43837 


64 


26 


•43837 


•44098 


•44359 


•44619 


•44879 


•45139 


•45399 


63 


27 


•45399 


•45658 


•45916 


•46174 


•46432 


•46690 


•46947 


62 


28 


•46947 


•47203 


•47460 


•47715 


•47971 


•48226 


•48480 


61 


29 


•48480 


•48735 


•48988 


•49242 


•49495 


•49747 


•50000 


60 


30 


•50000 


•50251 


•50502 


•50753 


•51004 


•51254 


•51503 


59 


31 


•51503 


•51752 


•52001 


•52249 


•52497 


•52745 


•52991 


58 


32 


•52991 


•53238 


•53184 


•53729 


•53975 


•54219 


•54463 


57 


33 


•54463 


•54707 


•54950 


•55193 


•55436 


•55677 


•55919 


56 


34 


•55919 


•56160 


•56400 


•56640 


•56880 


•57119 


•57357 


56 


35 


•57357 


•57595 


•57833 


•58070 


•58306 


•58542 


•58778 


54 


36 


•58778 


•59013 ; 


•59248 


•59482 


•59715 


•59948 


•60181 


53 


37 


•60181 


•60413 ! 


•60645 


•60876 


•61106 


•61336 


•61566 


52 


38 


•61566 


•61795 ! 


•62023 


•62251 


•62478 


•62705 


•62932 


51 


39 


•62932 


•63157 


•63383 


•63607 


•63832 


•64055 


•64278 


50 


40 


•64278 


•64501 


•64723 1 


•64944 


•65165 


•65386 


•65605 


49 


41 


•65605 


•65825 


•66043 


•66262 


•66479 


•66696 j 


•66913 


48 


42 


•66913 


•67128 


•67344 


•67559 


•67773 


•67986 


•68199 


47 


43 


•68199 


•68412 


.68642 j 


•68835 


•69046 ! 


•69256 


•69465 


46 


44 


•69465 


•69674 


•69SS3 ; 


•70090 


•70298 


•70504 


•70710 


45 





60' 


50' I 


40' 


30' 


20' 


10' 


0' 


Deg 






Natn 


ral Cos 


ine. 







Triuoxometrt. 



1C9 



Natural Sine* 



I>eg. 


0' 


10' 


20' 


30' 


40' 


50' 


60' 




45 


} -70710 


•70916 


•71120 


•71325 


•71523 


•71731 


•71933 


44 


46 


1 -71933 


•72135 


•72336 


•72537 


•72737 


•72936 


•73135 


43 


47 


•73135 


•73333 


•73530 


•73727 


•73923 


•74119 


•74314 


42 


43 


. T4314 


•74508 


•74702 


•74895 


•75083 


•75279 


1 -75470 


41 


49 


'] -75470 


•75661 


•75851 


.76040 


•76229 


•76417 


•76604 


40 


50 


i -76604 


•76791 


•76977 


.77162 


•77347 


•77531 


! -77714 


39 


51 


I -77734 


•77S97 


•78079 


.78260 


•78441 


•73621 


i -7SS01 


38 


52 


•78801 


•78979 


•79157 


•79335 


•79512 


•79688 


•79863 


37 


53 


•79863 


•80033 


•80212 


.80385 


•S0558 


•80730 


•80901 


36 


54 


•80901 


•81072 


•81242 


.81411 


•81580 


•81748 


•81915 


35 


55 


•81915 


•82081 


•82247 


.82412 


•82577 


•82740 


•82903 


34 


56 


•82903 


•S3066 


•83227 


.S3333 


•83543 


•83708 


•83867 


33 


57 


•83867 


•84025 


•84182 


.34339 


•84495 


•84650 


•84304 


32 


58 


•84S04 


•84958 


•85111 


.85264 


•85415 


S5566 


•85716 


31 


59 


•85716 


•85866 


•S6014 


.86162 


•86310 


•86456 


•86602 


30 


60 


•S6602* 


•86747 


•86891 


.87035 


•8717S 


•87320 


•S7461 


29 


61 


•87461 


•S7602 


•87742 


-87S81 


•38020 


•88157 


•SS294 


28 


62 


•SS294 


•88430 


•88566 


.83701 


•8SS35 


•8S968 


•89100 


27 


63 


•S9100 


•S9232 


•89363 


.89493 


•39622 


•89751 


•89879 


2Q 


64 


•S9879 


•90006 


•90132 


•9025S 


•903S3 


•90507 


•90630 


25 


65 


•90630 


•90753 


•90875 


•9099o 


•91116 


•91235 


•91354 


24 


66 


•91354 


•91472 


•91589 


.91706 


•91S11 


•9] 936 


•92050 


23 


67 


•92050 


•92163 


•92276 


•92387 


•92493 


•92609 


•92713 


22 


63 


•92718 


•92826 


•92934 


•93041 


•93147 


•93253 


•93358 


21 


69 


•9335S 


•93461 


•93564 


•93667 


•93768 


•93369 


•93969 


20 


70 


•93969 


•9406S 


•94166 


•94264 


•94360 


•94456 


•94551 


19 


71 


•94551 


•94646 


•94739 


•94332 


•94924 


•95015 


•95105 


IS 


72 


•95105 


•95195 


•95283 


•95371 


•95453 


•95545 


•95630; 


17 


73 


•95630 


•95715 


•95798 


•95831 


•95964 


•96045 


•96126' 


16 


74 


•96126 


•96205 


•96284 


•96363 


•96440 


•96516 


•965921 


15 


75 | 


•96592 


•96667 


•96741 


•96314 


•96S87 


•96953 


•97029' 


14 


76 


•97029 


•97099 


•97168 


•97236 


•97304 


•97371 


•97437; 


13 


77 


•97437 


•97402 


•97566 


•97629 


•97692 


•97753 


•97814: 


12 


73 


"97814 


•97874 


•97934 


•97992 


•93050 


•93106 


•98162: 


11 


79 


•98162 


•98217 


•98272 


•9S325 


•98373 


•9S429 


•98480' 


10 


SO 


•98480 


•93530 


•98530 


•9S628 


•98676 


•9S722 


•9S768 : 


9 


81 


•98768 


•98813 


•98858 


•93901 


•9S944 


•9S985 


•99026, 


8 


82 


•99026 


•99066 


•99106 


•99144 


•99182 


•99218 


•99254 ! 


7 


83 


•99254 


•99289 


•99323 


•99357 ; 


•993S9 


•99421 


•99452 


6 


84 


•99452 


•99482 


•99511 


•99539 


•99567 


•99593 


•99619 


5 


85 


•99619 


•99644 


•99668 


•99691 


•99714 


•99735 


•99756 


4 


86 


•99756 


•99776 


•99795 


•99813 


•99S30 


•99847 


•99862: 


3 


87 


•99862 


•99377 


•99891 


•99904 


•99917 


•99928 


•99939! 


2 


SS 


•99939 


•99948 


•99957 


•99965 


•99972 


•99979 


•99984 


1 


89 


•99984 


•99989 


•99993 


•99996 , 


•99998 


•99999 


1-0000 







60' 


50' 


40' 


30' 


20' ; 


10' 


0' 1 


Deg. 








Natii 


ral Cos 


ine. 







130 






Trkjonometry. 








Deg. 






Natural Tangent. 






0' 


1 10' 


20' 


30' 


40' 


50' 


60' 


' 





•ooooo 


•00290 


•00581 


•00872 


•01163 


•01454 


•01745 


89 


1 


•01745 


•02036 


•02327 


•02618 


•02909 


•03200 


•03492 


88 


2 


•03492 


•03783 


•04074 


•04366 


•04657 


•04949 


•65240 


87 


3 


•05240 


•05532 


•05824 


•06116 


•06408 


•06700 


•06992 


86 


4 


•06992 


•07285 


•07577 


•07870 


•08162 


•08455 


•08748 


85 


5 


•08748 


•09042 


•09335 


•09628 


•09922 


•10216 


•10510 


84 


6 


•10510 


•10804 


•11098 


•11393 


•11688 


•11983 


•12278 


83 


7 


•12278 


•12573 


•12869 


•13165 


•13461 


•13757 


•14054 


82 


8 


•14054 


•14350 


•14647 


•14945 


•15242 


•15540 


•15S38 


81 


9 


•15838 


•16136 


•16435 


•16734 


•17033 


•17332 


•17632 


80 


10 


•17632 


•17932 


•18233 


•18533 


•18834 


•19136 


•19438 


79 


11 


•19438 


•19740 


•20042 


•20345 


•20648 


•20951 


•21255 


78 


12 


•21255 


•21559 


•21864 


•22169 


•22474 


•227S0 


•23086 


77 


13 


•23086 


•23393 


•23700 


•24207 


•24315 


•24624 


•24932 


76 


14 


•24932 


•25242 


•25551 


•25861 


•26172 


•26483 


•26794 


75 


15 


•26794 


•27106 


•27419 


•27732 


•28045 


•28359* 


•28674 


74 


16 


•28674 


•28989 


•29305 


•29621 


•29938 


•30255 


•30573 


73 


17 


•30573 


•30891 


•31210 


•31529 


•31849 


•32170 


•32491 


72 


18 


•32491 


•32813 


•33136 


•33459 


•33783 


•34107 


•34432 


71 


19 


•34432 


•34758 


•35084 


•35411 


•35739 


•36067 


•36397 


70 


20 


•36397 


•36726 


•37057 


•37388 


•37720 


•38053 


•38386 


69 


21 


•38386 


•38720 


•39055 


•39391 


♦39727 


•40064 


•40402 


68 


22 


•40402 


•40741 


•4L080 


•41421 


•41762 


•42104 


•42447 


67 


23 


•42447 


•42791 


•43135 


•434S1 


•43827 


•44174 


•44522 


66 


24 


•44522 


•44871 


•45221 


•45572 


•45924 


•46277 


•46630 


65 


25 


•46630 


•469S5 


•47340 


•47697 


•48055 


•48413 


•48773 


64 


26 


•48773 


•49133 


•49495 


•49858 


•50221 


•50586 


•50952 


63 


27 


•50952 


•51319 


•51687 


•52056 


•52426 


•52798 


•53170 


62 


28 


••53170 


•53544 


•53919 


•54295 


•54672 


•55051 


•55430 


61 


29 


•55430 


•55811 


•56193 


•56577 


•56961 


•57347 


•57735 


60 


30 


•57735 


•58123 


•58513 


•58904 


•59296 


•59690 


•60086 


59 


31 


•60086 


•60482 


•60880 


•61280 


•61680 


•62083 


•62486 


58 


32 


•62486 


•62892 


•63298 


•63707 


•64116 


•64527 


•64940 


57 


33 


•64940 


•65355 


•65771 


•661S8 


•66607 


•67028 


•67450 


56 


34 


•67450 


•67874 


•68300 


•68728 


•69157 


•69588 


•70020 


55 


35 


•70020 


•70455 


•70S91 


•71329 


•71769 


•72210 


•72654 


54 


36 


•72654 


•73099 


•73546 


•73996 


•74447 


•74900 


•75355 


53 


37 


•75355 


•75S12 


•76271 


•76732 


•77195 


•77661 


•7812S 


52 


38 


•78128 


•78598 


•79069 


•79543 


•80019 


•80497 


•80978 


51 


39 


•80978 


•81461 


•81946 


•82433 


•82923 


•83415 


•S3909 


50 


40 


•83909 


•84406 


•84906 


•85408 


•85912 


•86414 


•86928 


49 


41 


•86928 


•87440 


•87955 


•88472 


•88992 


•89515 


•90040 


48 


42 


•90040 


•90568 


•91099 


•91633 


•92169 


•92704 


•93251 


47 


43 


•93251 


•93 79 6 


•94345 


•94896 


•95450 


•96008 


•96568! 


46 


U 


•9656S 


•97532 


•97699 


•98269 


•98843 


•99419 


l-OOOOi 


45 




60' 


50' 


40' 


30' 


20' 


10' 


0' 

1 


Deg. ' 








Natura 


?1 Cotangent* 













Trigonometry. 






131 


Natural Tangent* 


Deg. 


1 0' 


10' 


20' 


30' j 40' 


50' 


60' 


1 


45 


1-0000 


1-0058 


1-0117 


1-0176 


1-0235 


1-0295 


1-0355 


44 


46 


1-0355 


1-0415 


1-0476 


1-0537 


1-0599 


1-0661 


1-0723 


43 


47 


1-0723 


1-0786 


1-0849 


1-0913 


1-0977 


1-1041 


1-1106 


42 


48 


1-1106 


1-1171 


1-1236 


1-1302 


1-1369 


1-1436 


1-1503 


41 


49 


1-1503 


1-1571 


1-1639 


1-1708 


1-1777 


1-1847 


1-1917 


40 


50 


1-1917 


1-19S8 


1-2059 


1-2130 


1-2203 


1-2275 


1-2348 


39 


51 


1-2348 


1-2422 


1-2496 


1-2571 


1-2647 


1-2722 


1-2799 


38 


52 


1-2799 


1-2876 


1-2954 


1-3032 


1-3111 


1-3190 


1-3270 


37 


53 


1-3270 


1-3351 


1-3432 


1-3514 


1-3596 


1-3679 


1-3763 


36 


54 


1-3763 


1-3848 


1-3933 


1-4019 


1-4106 


1-4193 


1-4281 


35 


55 


1-4281 


1-4370 


1-4459 


1-4550 


1-4641 


1-4732 


1-4825 


34 


56 


1-4825 


1-4919 


1-5013 


1-5108 


1-5204 


1-5301 


1-5398 


33 


57 


1-5398 


1-5497 


1-5596 


1-5696 


1-5798 


1-5900 


1-6003 


32 


58 


1-6003 


1-6107 


1-6212 


1-6318 


1-6425 


1-6533 


1-6642 


31 


59 


1 6612 


1-6752 


1-6864 


1-6976 


1-7090 


1-7204 


1-7320 


30 


60 


1-7320 


1-7437 


1-7555 


1-7674 


1-7795 


1-7917 


1-S040 


29 


61 


1-8040 


1-8164 


1-8290 


1-8417 


1-8546 


1-8676 


1-8807 


28 


62 


1-8807 


1-8939 


1-9074 


1-9209 


1-9347 


1-9485 


1-9626 


27 


63 


1-9626 


1-9768 


1-9911 


2-0056 


2-0203 


2-0352 


2-0503 


2Q 


64 


2-0503 


2-0655 


2-0809 


2-0965 


2-1123 


2-1283 


2-1445 


25 


65 


2-1445 


2-160S 


2-1774 


2-1942 


2-2113 


2-2285 


2-2460 


24 


GQ 


2-2460 


2-2637 


2-2816 


2-2998 


2-3182 


2-3369 


2-3558 


23 


67 


2-3558 


2-3750 


2-3944 


2-4142 


2-4342 


2-4545 


2-4750 


22 


68 


2-4750 


2-4959 


2-5171 


2-5386 


2-5604 


2-5826 


2-6050 


21 


69 


2-6050 


2-6279 


2-6510 


2-6746 


2-69S5 


2-7228 


2-7474 


20 


70 


2-7474 


2-7725 


2-7980 


2-8239 


2-8502 


2-8769 


2-9042 


19 


7L 


2-9042 


2-9318 


2-9600 


2-9886 


3-0178 


3-0474 


3-0776 


IS 


72 


3-0776 


3-1084 


3-1397 


3-1715 


3-2040 


3-2371 


3-2708 


17 


73 


3-2708 


3-3052 


3-3402 


3-3759 


3-4123 


3-4495 


3-4S74 


16 


►74 


3-4874 


3-5260 


3-5655 


3-6058 


3-6470 


3-6890 


3-7320 


15 


75 


3-7320 


3-7759 


3-8208 


3-8667 


3-9136 


3-9616 


4-0107 


14 


76 


4-0107 


4-0610 


4-1125 


4-1652 


4-2193 


4-2747 


4-3314 


13 


77 


4-3314 


4-3896 


4-4494 


4-5107 


4-5736 


4-6382 


4-7046 


12 


78 


4-7046 


4-7728 


4-8430 


4-9151 


4-9894 


5-0658 


5-1445 


11 


79 


5-1445 


5-2256 


5-3092 


5-3955 


5-4845 


5-5763 


5-6712 


10 


80 


5-6712 


5-7693 


5-8708 


5-9757 


6-0844 


6-1970 


6-3137 


9 


81 


6-3137 


6-4348 


6-5605 


6-6011 


6-8269 


6-96S2 


7'1153 ! 


8 


82 


7-1153 


7-2687 


7-4287 


7*5957 


7-7703 


7-9530 


8-1443! 


7 


83 


8-1443 


8-3449 


8-5555 


8-7768 


9-0098 


9-2553 


9-5143 


6 


84 


9-5143 


9-7881 


10-078 


10-385 


10-711 


11-059 


11-430: 


5 


85 


11-430 


11-826 


12-250 


12-760 


13-196 


13-726 


14-300 


4 


86 


14-300 


14-924 


15-604 


16-349 


17-169 


18-074 


19-081 


3 


87 


19-081 


20-205 


21-470 


22-003 ! 


24-541 


26-431 


28*636 


2 


88 


28-636 


31-241 


34-367 38-188 


42-964 


49-103 


57*289 


1 


89 


57-289 


68-750 


85-939 


114-58 


171-88 ; 


343-77 


CO 








60' 


50' 

1 


40' 


30' 


20' 


10' 


0' 

1 


Dog. 


Natural Cotangent. 



132 






Trigonometi 


r. 














Natural Secant* 





Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


60' 







1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0001 


1.0001 


89 


1 


1.0001 


1.0002 


1.0002 


1.0003 


1.0004 


1.0005 


1.0006 


88 


2 


1.0006 


1.0007 


1.0008 


1.0009 


1.0010 


1.0012 


1.0013 


87 


3 


1.0013 


1.0015 


1.0016 


1.0018 


1.0020 


1.0022 


1.0024 


86 


4 


1.0024 


1.0026 


1.0028 


1.0031 


1.0033 


1.0035 


1.0038 


85 


5 


1.0038 


1.0040 


1.0043 


1.0046 


1.0049 


1.0052 


1.0055 


84 


6 


1.0055 


1.0058 


1.0061 


1.0064 


1.0068 


1.0071 


1.0075 


83 


7 


1.0075 


1.0078. 


1.0082 


1.0086 


1.0090 


1.0094 


1.0098 


82 


8 


1.0098 


1.0102 


1.0106 


1.0111 


1.0115 


1.0120 


1.0124 


81 


9 


1.0124 


1.0129 


1.0134 


1.0139 


1.0144 


1.0149 


1.0154 


80 


10 


1.0154 


1.0159 


1.0164 


1.0170 


1.0175 


1.0181 


1.0187 


79 


11 


1.0187 


1.0192 


1.0198 


1.0204 


1.0210 


1.0217 


1.0223 


78 


12 


1.0223 


1.0229 


1.0236 


1.0242 


1.0249 


1.0256 


1.0263 


77 


13 


1.0263 


1.0269 


1.0277 


10284 


1.0291 


1.0298 


1.0306 


76 


14 


1.0306 


1.0313 


1.0321 


1.0329 


1.0336 


1.0344 


1.0352 


75 


15 


1.0352 


1.0360 


1.0369 


1.0377 


1.0385 


1.0394 


1.0403 


74 


16 


1.0403 


1.0411 


1.0420 


1.0429 


1.0438 


1.0447 


1.0456 


73 


17 


1.0456 


1.0466 


1.0475 


1.0485 


1.0494 


1.0504 


1.0514 


72 


18 


1.0514 


1.0524 


1.0534 


1.0544 


1.0555 


1.0565 


1.0576 


71 


19 


1.0576 


1.0586 


1.0597 


1.0608 


1.0619 


1.0630 


1.0641 


70 


20 


1.0641 


1.0653 


1.0664 


1.0676 


1.0687 


1.0699 


1.0711 


69 


21 


1.0711 


1.0723 


1.0735 


1.0747 


1.0760 


1.0772 


1.0785 


6S 


22 


1.0785 


1.0798 


1.0810 


1.0823 


1.0337 


1.0850 


1.0S63 


67 


23 


1.0863 


1.0877 


1.0890 


1.0904 


1.0918 


1.0932 


1.0946 


66 


24 


1.0946 


1.0960 


1.0974 


1.0989 


1.1004 


1.1018 


1.1033 


65 


25 


1.1033 


1.1048 


1.1063 


1.1079 


1.1094 


1.1110 


1.1126 


64 


20 


1.1126 


1.1141 


1.1157 


1.1174 


1.1190 


1.1206 


1.1223 


63 


27 


1.1223 


1.1239 


1.1256 


1.1273 


1.1290 


1.1308 


1.1325 


62 


2S 


1.1325 


1.1343 


1.1361 


1.1378 


1.1396 


1.1415 


1.1433 


61 


29 


1.1433 


1.1452 


1.1470 


1.1489 


1.1508 


1.1527 


1.1547 


60* 


30 


1.1547 


1.1566 


1.1586 


1.1605 


1.1625 


1.1646 


1.1666 


59 


31 


1.1666 


1.1686 


1.1707 


1.1728 


1.1749 


1.1770 


1.1791 


58 


32 


1.1791 


1.1833 


1.1835 


1.1856 


1.1878 


1.1901 


1.1923 


57 


33 


1.1923 


1.1946 


1.1969 


1.1992 


1.2015 


1.2038 


1.2062 


56 


34 


1.2062 


1.2085 v 


1.2109 


1.2134 


1.2158 


1.2182 


1.2207 


55 


35 


1.2207 


1.2232 


1.2257 


1.2283 


1.2308 


1.2334 


1.2360 


54 


36 


1.2360 


1.2386 


1.2413 


1.2440 


1.2466 


1.2494 


1.2521 


53 


37 


1.2521 


1.2548 


1.2576 


1.2604 


1.2632 


1.2661 


1.2690 


52 


38 


1.2690 


1.2719 


1.2748 


1.2777 


1.2807 


1.2837 


1.2867 


51 


39 


1.2867 


1.2898 


1.2928 


1.2959 


1.2990 


1.3022 


1.3054 


50 


40 


1.3054 


1.3086 


1.3118 


1.3150 


.1.3183 


1.3216 


1.3250 


49 


41 


1.3250 


1.3283 


1.3317 


1.3351 


1.3386 


1.3421 


1.3456 


48 


42 


1.3456 


1.3191 


1.3527 


1.3563 


1.3599 


1.3636 


1.3673 


47 


43 


1.3673 


1.3710 


1.3748 


1.37S5 


1.3824 


1.3862 


1.3901 


46 


44 


1.3901 


1.3940 


1.3980 


1.4020 


1.4060 


1.4101 


1.4142 


45 




60' 


50' 


40' 


30' 


20' 


10' 


0' 


Deg. 








N al ui 


al Cose 


cant. 


- 











Trigonometry. 






133 


Beg. 


Natural Secant* 


0' 


10' 


20' 


30' 


1 40' 


50' 


60' 


i 

! 


45 


1-4142 


1-4183 


1-4225 


1-4267 


i 1-4309 


1-4352 


1-4395 


1 '44 


46 


1-4395 


1-4439 


1-4483 


1-4527 


1-4572 


1-4617 


1-4662 


l 43 


47 


1-4662 


1-4708 


1-4755 


i 1-4801 


1-4849 


1-4896 


1-4944 


42 


48 


1-4944 


1-4993 


1-5042 


1-5091 


1-5141 


! 1-5191 


1-5242 


41 


49 


1-5242 


1-5293 


1-5345 


1-5397 


1-5450 


1-5503 


1-5557 


40 


50 


1*5557 


1-5611 


1-5666 


1-5721 


1-5777 


1-5833 


1-5890 


39 


51 


1-5890 


1-5947 


1-6005 


j 1-6063 


1-6122 


1-6182 


1-6242 


38 


52 


1-6242 


1-6303 


1-6364 


1-6426 


1-6489 


1-6552 


1-6616 


37 


53 


1-6616 


1-6680 


1-6745 


1-6811 


1-6878 


1-6945 


1-7013 


36 


54 


1-7013 


1-7081 


1-7150 


1-7220 


1-7291 


1-7362 


1-7434 


35 


55 


1-7434 


1-7507 


1-7580 


1-7655 


1-7730 


1-7806 


1-7882 


34 


56 


1-7SS2 


1-7960 


1-8038 


1-8118 


1-8198 


1-8278 


1-8360 


33 


57 


1*8360 


1-8443 


1-8527 


1-8611 


1-8697 


1-8783 


1-8870 


32 


58 


1-8870 


1-8959 


1-9048 


1-9138 


1-9230 


1-9322 


1-9416 


31 


59 


1-9416 


1-9510 


1-9606 


1-9702 


1-9800 


1-9899 


2-0000 


30 


60 


2-0000 


2-0101 


2-0203 


2-0307 


2-0412 


2-0519 


2-0626 


29 


61 


2-0626 


2-0735 


2-0845 


2-0957 


2-1070 


2-1184 


2-1300 


2S 


62 


2-1300 


2-1117 


2-1536 


2-1656 


2-1778 


2-1901 


2-2026 


27 


63 


2-2026 


2-2153 


2-2281 


2-2411 


2-2543 


2-2676 


2-2811 


26 


64 


2 2811 


2-294S 


2-3087 


2-3228 


2-3370 


2-3515 


2-S662 


25 


65 


2.3662 


2-3810 


2-3961 


2-4114 


2-4269 


2-4426 


2-4585 


24 


66 


2*4585 


2-4747 


2-4911 


2-5073 


2-5247 


2-5418 


2-5593 


23 


67 


2-5593 


2-5769 


25949 


2-6131 


2-6316 


2-6503 


2-6694 


22 


68 


2-6691 


2-6SS8 


2-70S5 


2-7285 


2-7488 


2-7694 


2-7904 


21 


69 


2-7904 


2-3117 


2-8334 


2-8554 


2-8778 


2-9006 


2-923S 


20 


70 


2-9238 


2-9473 


2-9713 


2-9957 


3-0205 


3-0458 


3-0715 


19 


71 


3-0715 


3-0977 


3-1243 


3-1515 


3-1791 


3 2073 


3-2360 


IS 


72 


3*2360 


3-2653 


3-2951 


3-3255 


3-3564 


3-38S0 


3-4203 


17 


73 


3-4203 


3-4531 


3-4867 


3-5209 


3-5558 


3-5915 


3-6279 


16 


74 


3-6279 


3-6651 


3-7031 


3-7419 


3-7S16 


3-S222 


3-8637 


15 


75 


3-8637 


3-9061 


3-9495 


3-9939 


4-0393 


4-0859 


4-1335 


14 


76 


4-1335 


4-1823 


4-2323 


4-2836 


4-3362 


4-3900 


4-4454 


13 


77 


4-4454 


4-5021 


4-5604 


4-6202 


4-6S16 , 


4-7443 


4-S097 


12 


78 


4-8097 


4-8764 


4-9451 


5-0158 


5.0SS6 1 


5-1635 


5-2408 


11 


79 


5-2408 


5-3204 


5-4026 


5-4874 


5.5749 


5-6653 


5-75871 


10 


SO 


5-7587 


5-8553 


5-9553 


6-0588 


6-1660 


6-2771 


6*3924 


9 


81 


6-3924 


6-5120 


6-6363 


6-7654 


6-S997 


7-0396 


7-1852: 


8 


82 


7-1852 l 


7-3371 


7-4957 


7-6612 


7-8344 


8-0156 


8-2055 ; 


7 


83 


8-2055 


8-4046 


8-6137 


8-8336 


9-0651 


9-3091 


9-5667 


6 


84 


9-5667 


9-8391 


10-127 i 


10.437 i 


10-758 


11-104 


11-473 


5 


S5 


11-473 


11-868 


12-291 1 


12.745 


13-234 


13-763 


14-335 


4 


86 


14-335 


14-957 


15-636 


16.380 


17-198 


18-102 


19-107 


3 


87 


19-107 


20-230 


21-493 


22.925 


24-562 26-450 


28'653 ; 


2 


88 


28-653 


31-257 


34-3S2 38.201 


42-975 49-114 


57*298, 


1 


89 


57-298 68-757 


S5-945 114.59 171. S3 343-77 


00 i 





\ 


60' 


50' 


40' ; 30' j 


20' 10' 

I 


r j 


Beg. 


Natural Cosecant* 



134 



Right-Ancled Triangle. 



^3 



FORMULA FOR RIGHT-ANGLED TRIANGLES. 
136 



c 



a = 



sin. C 

b 
cos.C 



V si 



Q 



sin.2C' 



b = a cos. C, 
£ = c cot. C, 
6 = a sin.I?, 
6 = c tan.P, 

V tan. 



C' 



a a sin.2C 



sin. C»-, 
a 



cos.C= — , 



tan. C = -r , 

4Q 
sin.2C = -;, 

^ 20 
tan.C = -^-, 



10, 

11, 



Q=Jc*cot.C, 12, 



Q-Jc>/(fi+c)(a — c) 13, 



14, 
15, 
16, 
17, 

18, 



Say the angle to be O — 60°. In the first column of the table of sines, 60° 
corresponds with 0*86602 in the next column, which ' is the length of sin. 60°, 
when the radius of the circle is one, or the unit, and the expression sin. 60°X36 
means 0*86602X36 = 31-17672, and likewise with all the other Trigonometrical 
expressions. 

In a triangle the functions for an angle have a certain relation to the oppo- 
site side ; it is this relationship which enables us to solve the triangle by the ap- 
plication of Simple Arithmetic. 

In triangles the sides are denoted by the letters a, &, and c ; their respective 
opposite angles are denoted by A, B, and C, and the area by Q. 

Example 1. Fig. 136 The side c in a right angled Triangle being 365 'feet, and 
the angle C = 39° 20'. How long is the side a = ? 

= SM? = sin. 390.20' = 0^63383 = 575 ' 86 feet > the anSWer * 



Formula 2. 



Oblique-angled Triangle. 



FORMULA FOR OBLIQUE-ANGLED TRIANGLES. 

137 ^ w 138 





a: b = sin. A : sin.2?, and 5 : c = sin.B : sin.C. 
a : c = sin. A : sin.C, and Q : ah = sin.C : 2. 






c sin. A 
sin.C ■ 

c sin.A 
: sin.(A-hB)' 

2Q 

b sin.C 



sin.C ' 

2Q 
c sin. A* 



. r, c sin.B 

sin. C = 7 — , 

b 



sin.C = 



c sin. A 



sin.A = Tc , 



sin.A ■■ 



2Q 

be 9 

a sin. C 



5 = i(a+b+c) 12, 



<7 =• V #*+c a — 26 c cos. A, 10 3 



/ 2 Q sin.A. ,.. 

a ~ V sin.£sin.(A+J5) ii! 



sin.U^>V5^^ c) ,13, 

8in.lB-v/IE^z5l4, 

V ao* 



cofl-u-y/'t'-".}, 15, 
cos.4B=* AEE^J, 16, 

V "0 

be sin.A 



Q = 



2 ' 
a& sin. C 



17, 
18, 



n c* sin.A sin.jB ■, Q 
W= 2sin.(A+£) ' ly ' 

Q=V r S-a)(S-i)(S-c) 20, 



,_ /2Qsin.(^-C) 21 

V sin. J. sin.C 



2Q sin.C 9 2 
sin.A sin.(A+C) 



136 



Spherical Trigonometry. 



To Solve Triangles Mechanically # 

PLATE III. 

The ac2ompanying diagram is so constructed that the moveable arm repre- 
sents the hypothenuse, the square lines the two sides, and the circular scale the 
angles in a right-angled triangle. The scale numbered from the centre, towards 
will he called b, and the one at right angle to he called c. 

Example 1. Let the lines that form the right-angle be given as b = 12 and 
c = 4 inches. Required the hypothenuse a, and the angles B and G? 

Find where the two lines 12 and 4 crosses each other, move the arm to this 
crossing-point, which then indicates the length of the hypothenuse — 12-65 on 
the arm a ; the two angles will be found at a on the scale. B = 71° 40' and 
C=18°20'. 

If one angle and a side is given, set the arm on the given angle, and the in- 
tersection of the given side with the arm shows the length of the hypothenuse 
and the other side. 

An oblique angled triangle can be two right angled triangles by drawing a 
line from the largest angle perpendicular to the opposite side, and can be solved 
by this diagram. 

Example 2. An oblique angled triangle being a = 65 feet, C= 34° 30' and 
B = 68° 20'. 

Required the two sides b and c ? 

Set the arm on the given angle 34° 30', and at 65 feet on the arm will be 
found the height of the triangle = 37 feet on scale c, and one part of the side b 
is 54 feet on the scale b. 

From the given angle B = 68° 20 7 

Subtract the complement of 34° 30' = 55° 30' 

Set the arm on the angle, 12° 50' 

Now at the height 37 on the scale b will be found 38 feet on the arm, which is 
the length of the side c, and the other part of the side b is 9 feet on the scale c, 
then b = 54+9 = 63 feet. 

In a similar manner any plane triangle can be so solved. By a little practice, 
this Table is very useful for approximating triangles. 



SPHERICAL TRIGONOMETRY. 

Splierical Trigonometry treats of triangles which are drawn (or ima- 
gined to be) on the surface of a sphere; their sides are arcs of the great circle 
of the sphere, and measures by the angle of the arc. Therefore the trigonome- 
trical functions bear quite a different relation to the sides. 

Every section of a sphere cut by a plane is a circle. A line drawn through the 
centre and at right angles to the sectional circle is called an axis, and the two 
points where the axis meets the surface of the sphere are called the poles of the 
sectional circle. 

X_ 139 

When the cutting plane goes through the 
centre of the sphere, it will pass through the 
great circle, and is then called the Equa- 
tor for the poles. Axis == N.S- Equator — 
G.E.T.W. 

Three great circle-planes, aa'a f 'a'", bb'b", 
and ceV, cutting a sphere, JVESW, will form 
a solid angle at the centre O, and a triangle 
ABC on the surface of the sphere, in which 
the arcs a, b, c, are the sides. The angles form- 
ed by each two planes are congruent to each 
of the appertinent angles A, B, and C, 




/'hilrlll 



2}/ hie for so /r in (j 





Right-Angled Spherical Triangle 



RIGHT-ANGLED SPHERICAL TRIANGLE. 

140 




sin. 6 = 
tan.c = 
cot.C- 
tan.c = 
cos. a = 
cos. I? = 

tan. a = 

sin.c = 

sin. a = 

sin. C = 



sin.a sin.I?, 
tan.a cos.B, 
cos. a tan.JS, 
sin. b tan.C, 
cos.£ cos.c, 
cos.& sin.C, 
tan. b 
taSC' 




9, 
10, 
11, 



sin.f? = 



sin. C -■ 



_ sin. b 
sin. a' 

tan. b 
tan. a 



. rt tan.c 

tan. C = - — - , 

sin. 6 



tan.i? = 



cos.c = 



cos. b -■ 



cos.a = 



tan. b 
sin.c' 

cos.C 
sm7B' 

cos.I? 

sinTC" 

cot.C 
tan.£' 



12, 
13, 
14, 
15, 
16,' 
17, 
18. 



The sum of the three angles in a spherical triangle is greater than two right 
angles, and less than six right angles. 

By Spherical Trigonometry we ascertain distances and courses on the surface 
of the earth ; positions and motions of the heavenly bodies, &c, &c. Examples 
will be furnished in Geography and Astronomy. 

Example 1. Fig. 140 In a right-angled spherical triangle the side or hypothe- 
nuse a = 36° 2C, the angle B = 68° 50'. How long is the side 5 = 1 

Formula 1. sin.o = sin.a.sin.i? = sin.36°20 / Xsin.6S°60 / . 

a log.sin. 36° 20 7 = 1:77267 

B log.sin. 68° 50' = 1:96966 

The answer, log.sin. 33° 32' = 1:74233 or 6 = 33° 32'. 



12* 



Oblique-Angled Spherical Triangle. 




sm.a : sin.i = sin. J. : sin.I>, 
sin.6 : sin.c = sin.5 : sin.C, 



sm.a ■■ 



sin.b-. 



sin.fl sin.A 
sin.£ ' 

sin.c sin.J9 
" sin. ' 



Un.(a+b) = tan.|c cos.^^% - 

, . , x , sin.^M— -I?) 

v y sin.i(A + 2?) 

tan. 4 (^) = cot.U C ^f^, 
tan.M^-^) = cot.U^§=^, 

cot.U = ton.i(£ - C) t n w ( / +C) > 

■* ^ v J sm.i{£> — c) 

. , . , , ,> sin.j Q4+.B ) 
tan.ic = tan.i(«- S) sha(A _2jy 



19 

20, 

- 21, 
22, 

- 23, 
24, 

- 25, 
26, 



Example 2. Fig. 141 bliq ue angled spherical triangle, c = 72° 30'. B = 17° 30 7 . 
C =79° 50'. 
How long is the side & = ? 

_ ^ . , sin.csin.JB sin.72° S0'Xsin.l7° 30* 
^^ 20. sm.6 = — - gi5 ^- _ sin . 79 o 5Q , 



The answer 



+ log.sin. 72° 30' = 1:97942 
+ log.sin. 17° 30' = 1:47812 
-f = 11:45754 

4- log. sin. 79° 50' = 1:99312 
log.sin. 16° bV = 1:46442 



or 6= 16° 56/ 



Oblique Angled Spherical Triangle. 



139 



OBLIQUE-ANGLED SPHERICAL TRIANGLE. 
142 




tan.i(m-f7i)tani(m — n = tan.i(a+c)tanj(a — c) 
tan.m = tan.c cos. A, - 27, 



tan.C=! in i W2tai1 ^:, 
sin.(&— m) 



cos.a — 



cos.c cos. (b — m) 



cos. ft = 



cos.m 
cos.a cos.TTi 



cot.m — 



cos.c 
b = m±n. 
cos.c tan. A 



tan. a 
5 = a+b+c S = J.+^+C, 



sin. J J. : 



•*-V 



sin. (5 — c) sin. (5 — 6) 
sin.6 sin.c 



cos.ff cos.(S — A) 
§m.B sin. C ' 



- 28, 
29, 

- 30, 

31, 

- 32, 

33, 



To Find the Area of a Spherical Triangle. 

Let Q he the area of the triangle in square degrees ; if H = radius of the 
sphere, the length of one degree will he, 

2ftR , i? a 

" "360" ' ° r ° ne SqLUare degre8 " 3285*8' 



cot.JQ = 
sin.JQ = 



cot.Jc cot.ia+cos.^ 
smJ? v> 

sin.ic &m.\a sin.J5 



cos.p 



140 



Conic Sections. 



CONIC SECTIONS. 

A Conic Section is the section obtained when a plane cuts a cone. 

The conic sections are of five different kinds, namely. 

1st. Triangle. "When the plane cuts the cone through its axis. 

2d. Circle. When the plane cuts the cone at right angles to its axis. 

3d. Ellipse. When the plane cuts the cone obliquely passing through the two 
sides. 

4th. Parabola. When the plane cuts the cone parallel to one side. 

5th. Hyperbola. When the plane cute the cone at an angle to the axis less than 
the angle of the axis and the side of the cone. 

The position of a point in a plane surface is determined by its course and dis- 
tance from a given point on a straight line, or by its distances from two lines 
inclined to one another 

Let AB and CD be two infinite lines inclined to 
one another, and P being a point in the same plane 
as the lines. It is evident that the position of this 
point P is determined by the distances x and y from 
the lines AB and CD. Those lines and distances 
are called, AB the axis of ordinate, and CD the 
axis of abscissa, y the ordinate, x the abscissa, and 
o the origin. 

The abscissa x is commonly taken on the absciss' axis. Now take different 
values of the abscissa x, and by some formula or rule calculate the ordinate y ; 
then a number of points P^P", &c, may be obtained; join those points by a 
line, then the rule or formula is called an equation for that line. Equations 
of this kind will here be furnished for the curve in the conic sections. 

Transverse-axis is the longest line that can be drawn in an Ellipse. 

Conjugate-axis is a Mne drawn through the centre, at right angles to the 
transverse axis. 

Parameter of any diameter is a third proportional to that diameter, and its 
conjugate. 

Fociis is the point in the axis where the ordinate is equal to half the di- 
ameter. 





144. 



Cycloid. 



y = 0-637 Vx(nd— *), 
e = l-2lld, p-0-M7d. 



Conic Sections. 




a : b = sin.w : sin.v, 
x : c = sm.w : sin.z, 
x : d = sin.w; : sin. (2+1;), 



b = 



d = 



a sm.v 

sin.it; ' 

x sin.z 
# sin.(2+w) 



sin.w 



y* = c(d+Z>). £ee Fig. 64, page. 91, 



y* = 



x.sm.z ,a sin.fz+v) a sin.tA 



sm.w; 



sin.w ' 



x.&m.z, . r _ x 

V = —. — 5- (a sin.[2+fl J + a sm.v). 
* sm.W »- j / 



This is the general formula for all conic sections. 

In any conic section, a point P can be calculated by this formula 6, but for 
the different sections, it will be found greatly simplified on the'next pages. 
For a Parabola z+v = 180, therefore sin.z = sin.v, and 

„ o x sin. 3 y 

y — — -. -. 

sm. 2 w 



142 



Conic Sections. 



^*-* / 


p^> 




V 


*\ 


t * ■ 






tf «2> 


-* 





146. Circle, 

y = >/2r# — a? 2 , 
y*-f* 2 



2* ' 



«=r+vr 2 — y\ 




147. 



Circle. 



y = V r a — 5% 



# = Vr* 1 — /, 




X^ 



148. 



Circte -Arc. 



3/ = Va 2 +cr — j? 2 
c 2 — 4A a 



149. 



C7rc/e -Arc. 







c» — 46* 
8A # 



150. 



ElEpse. 








151. 



Ellipse. 



\/^G)' 



Conic Sections. 



143 




152. 



Ellipse. 



e = jrrf — n*, 
2n* 



'153. 



Ellipse. 






A \\ 


*J 


X \ * V 








N A 



i2 = 2m— r, 
r = 2m — .R. 



154. Parabola, 

y = \/~/?i, j9 = 4m, 



r = \fy*-i(x — m) a = #-f m, 
3/= \A* a — 4a?V 



?/= 2 V #(#-}-m) — #', 



2 = 2\/#(<z+m). 




Hyperbola. 



R — r = 2m, e = 



V = — \/ (e a — l)(.ra — m a ), 
^ m 



e » 1 



-V 



(.r-fm) 2 



-1. 



144 Mechanics.— Statics. 



MECHANICS. 

Mechanics is that branch of Natural Philosophy which treats of the action 
of force, motion, and power. Mechanics is divided into four parts, namely, 
Statics the science of forces in equilibrium. 

Dynamics, the science of forces in motion, it produces power or effect. 
Hydrostatics, the science of fluids in equilibrium 
Hydrodynamics the science of fluids in motion, its causes, power or effects. 

Statics* Le ver* Momentum* 

Lever is an inflexible bar, supported in one point called the Fulcrum, or, 
centre of motion. The length of a lever is measured from the fulcrum to where 
the force or resistance acts, (when the force acts at right angles to the lever) or, 
the length of a lever is measured from the fulcrum at right angles to the direc- 
tion of the force. 

W= Weight, and I = lever for W\ ~ a v . ' 
F = Force, and L = lever for f\ See Fl ^ 156 ' 

Momentum is the product of force or weight, multiplied by the length of 
the lever it acts upon. 

The products WT and FL are called Statics Momentums ; when these lno- 
mentums are equal there will be no motion, and the weight TTwill balance 
the force F. "When one momentum is greater than the other, there will be a 
motion, and the velocity of that motion is measured by the difference of the 
momentums. 

Levers are of three distinct kinds, with reference to the relative positions of 
the Force F, Weight W, and Fulcrum C. 

1st. Fulcrum C, is between the force F, and the weight W. 

2d. Weight W, is between the fulcrum C, and the/o?re F. 

3d. Force F, is between the fulcrum C, and the weight W. 

Example 1. Figure 156. The weight W= 68 pounds, the lever I = 3*86 feet, 
andi= 10 feet 6 inches. 

Required the force F= ? 

, . „ Wl 68X3-36 oe 
Formula 1. F= -=r - = ■ 1A - — = 25 pounds nearly. 

Jj LU'O 

a = distance between the force F and the weight W. 

The formula 3, 4, 7, 8, 11, 12, are for unding the fulcrum C, when the force 
F, weight W. and the distance a, are given. 

Example 2. Fig. 157. The force F= 360 pounds, W= 1870, and a -= 8 feet, 4 
inches. 

Required the position of the fulcrum c? 

t * 7 *h 360X8-333 2999-988 - Q . 

Formula!. 1= jyZTj?- 1870 ^ 3 6Q " TW~ 1986feet< 

L = 8*333+19-86 = 28-193 feet, the answer. 

Example 3. Fig.161.The weight of the lever is Q = 18 pounds. The centre of 
gravity is x = 2*25 feet from the fulcrum. W= 299 pounds, I = 5'5 feet, and 
/,= 11-95. 

Eequired the force F= ? in pounds. 

' WJ-9»_ 299X5-5-18X2-25 =134 . 25 ^^ 
L 31*95 

Inclined Plane* 

Example 4. Fig. 180. A load W= 3466 pounds, is to be drawn up an inclined 
plane, I = 638 feet long, and h = 86 feet high. 
What force is required to keep the load on the inclined plane ? 
hW 86X3466 
F= ~l~ =■ 638~~ = 4 6 7'2 pounds. 



Mechanics. — Statics. X45 



Example 4. Fig. 184. A Cylinder of cast iron, weighing W— 5245 pounds, is to 
be rolled up an inclined plane ; the angles v = 18° 20' and t/ = 8° 10 7 

What force is required to keep the cylinder on the plane? 

F = W. sin.(>H-tO =5245Xsin.(18° 20'+8° 10') = 2340 pounds. 

Example 5. Fig. 185. An iron hall which weighs 398 pounds, is tied to an in- 
clined plane with a rope ; the angle of the rope and the inclined plane is 
t/ = 16° 40', and v = 14° 30 7 . What force is acting on the rope ? 

, = Nu = 393X^30' = 1(H ^ 

cos.i/ cos.l6° 40 7 

Example 6. Fig. 170. What force F is required to raise a weight W= 8469 
pounds, by a double moveable pulley ? 

F= ±W= iX8469 = 2117-25 pounds. 
Example 7. Fig. 173. How much weight can a force F= 269 pounds lift by 
three compound moveable pulleys ? 

W= 2 U F = 2 S X269 = 2152 pounds, the answer. 

Screw. 

Example 8. Fig. 189. What force is required to lift a weight W= 16785 pounds, 
by a screw, with a pitch P = 0-125 feet, the lever being r = 5 feet. 4 inches ? 

_ WP 16785X0-125 MM , .. 

F=s -^T r " 2X3-14X5-333 " *** *™**> ** anSWer ' 
Including friction the force .Fwill be 

F _ W(P+fdn ) 

2 71 r 
Find the friction f on page 155. d diameter of the screw. 

Wedge. 

Example 9. Fig. 186. The head of the wedge a = 3 inches, and length 
I = 16£ inches ; the resistance to be separated is R = 4846 pounds. Required 
the force F = ? (Friction omitted.) 

4846X3 QQ1 , 

™ — lg g = 881 pounds. 
Ib'o 
Including friction the force F will be, 245 



*--[■*+'(•+£)] 



in which the friction / is to he found on page 155. 
Catenaria. 

Example 9. An iron chain 256 feet long, weighing 1560 pounds, is to be sus- 
pended between two points in the same horizontal line, but 196 feet apart. 

How deep will the chain hang under the line of suspension, and with what 
force will the chain act at the points of suspension ? 

Figure and Formula 178. we have given, 

W= JX156 = 780 poun ds, I = iX256 = 128 feet, and a = £X196 = 98 feet. 

h =x 0-6525 j/128» — 98* = 53-73 feet, the required depth under the horizontal 
line. 

2 V 53*73 
cot.v = •— - — = 1-096, or v = 41° 44', and 2v = 89° 28'. 

The required force will be, 

_ 780Xsin.44° 44' _, n 
** sin.89°28' = 649 P ° Und8 ' 
13 K _____= 



146 



Lever and Static Momentum, 



I x 



./££ 



156. F : W = Z : L, F L = Wl, 



F = 



Wl 

L ' 



W = 



FL 



I 
Wa 



c ~ w+f, * - w+y 



1,2, 

3,4, 



2d 



l > 



157. F: W=l:L, FL~ Wl, 



L = 



Wa 
W-F 



1 = 



Fa 



7,8, 



158. F: W=l: L, FL= W Z, 



S4 



A 






W7 



w. 



FL 



oTio, 



^=^V n ' 12 - 



^^ 




159. 

af+a'f+arf = 6r+6V+ 5 V. 

If the sum of the momen turns that act to more 
the hody in one direction are equal to the sum of 
the momentums that act opposite, the acting 
forces will be in equilibrium; c being the centre 
or fulcrum, 



CU X 




160. 

To find the fulcrum c when three forces act on 
one lever 

Rx= Q(a — ft — a?)+P(a — a?), 

_ Qa+Pa — Qb 
x - R+Q+P ' 




161. 

Q = weight of the lever, x = distance from the 
centre of gravity of the lever to the fulcrum. 
Balance the lever over a sharp edge, and the centre 
of gravity is found. 

„_ Wl -Qx w _ fl+Qx 



Lever and Static Momentum. 



147 




162. 
F- 



F:W=r:R. FR = Wr, 



Wr 



Wr 
R' R ~ F' 







163. 
F = 



Wi 



RR 



W = 



FRR' 

/ 1 

r r 



71 = number of revolutions of the wheels, 

n : ri = r' : R, v : v' = rr' : RR', 

V = velocity of W, v' = velocity of F* 




164. 
F = 



Wrr'r" 
R R'R" ' 



W = 



FRR'R" 
r r'r" 



n : n" = r'r" : R R', v : v' = rr'r" 
RR'R". 

r r'r" &C. = radii of the pinions. 
R R'R"&C. = radii of the wheels. 



s-Sp 




165. 

Let P and Q represent the magnitudes and direc- 
tions of two forces which act to move the body B. 
By completing the parallelogram, there will be ob- 
tained a diagonal force F, whose magnitude and di- 
rection is equal to the sum P and Q. J 7 * is called 
the resultant of P and Q. 




166. 

If three or more forces act in different directions 
to move a body J5, find the resultant of any two of 
them, and consider it as a single force. Between 
this and the next force find a second resultant, 
thus : P. Q, and Pare magnitudes and directions of 
the. forces. P+Q = r, r+R = F== P+ Q+R, or F 
is the magnitude and direction of the three forces, 
P, Q, and R. 




>£> 



a/ 



167. 



A force Q acting (alone) on the body P, can move 
it to a in a unit of time, another force P is able to 
move it to b in the same time ; now if the two 
forces act at the same time, they will move the 
body to c. c is the resultant of a and 6. 



143 



PULT^TS. 



k^— *r 



1168. 




H§£ 



iv^t 



Pullets. — A single fixed Pulley. 
F: W=R: R, or F = W, 
v = v f . 



169. 

A single moveable Pulley. 

F: W= R: 2R, or F=IW, 

If the force .Fbeing applied at a and act upwards, 
the result will be the same. 

v' - 2v. 





171, 



A double moveable Pulley. 

F: W=R:4:R, or F= IW, 

v' = 4t>. 

JL double moveable Pulley. 
F: W=R:±R, or F = {W, 



* 2u' 



o : t>' = 1 : 2m. 




172. 

Quadruple moveable Pulley. 
F=hW. F:W=R:H& 

Let w = any number of moveable pulleys, the 

w 



F = 



2u' 



v : v' = 1 : 2«. 



173. 



Compound Pulleys. 



U = number of moveable Pulleys. 
IF 

'"a* W = 2^, 

v : v' = 1 : 2 U . 



Funicular and Catenarian. 



149 




N^OT 



il74. 

An oblique fixed Pulley, 

j M F: W = 2cos.z: 1, 
F 




175. 



W: F = sin. (#+?;) : sin.z, 

p = W sin. a? /. = Wsin.v 

sin. (#+?;) ' sin.far+i'/ 

W:/~ sin.(arh>) : sin.v. 




17G. 



sin.^+t')' 



/= 7 



TV sin. w 



sin.(w-i-z) 




177. 

W-P+G+22+S, F=F',f=f, 

p _ TVsin.a? /._ T Tsin.tt 
~ sin.^^)' ^ sin.(#+i>)' 




178. 
I 



VT^. 



A =0-8660 \/> — a*, 



2 = length, and TF— weight of half the chain,^, 

cot. - ?*, F = ,^y,/ = Wtan.». 
a sm.2 K 




sm.2v 

cot.u = — . 
a 

TV = weight of half the numher of halls. 



IS* 



150 



Inclined Plane. 





180. 



F=Wh = Wsm.v, 



W= 



Fl 



Wl 
w = -— - = W cos.v. 
b 



181. 



F 
W = 

sin^iM-i/)' 

W = W cos.(vW). 



^^ 


*^ 


^tiK^ 




#K 




-^r ^ 





182. 



F = 



W sin.?; 
cos.u 

F cos.t/ 



sm.v 
= W (cos.iM-sin.v. tani/). 




183. 

To solve an Inclined Plane by diagrams. 

F = magnitude and direction of the 
force, which is obtained by completing 
i ! the parallelogram. 

By calculation see Formula, Fig. 180. 



4 VF 



W = weight of the body, and direc- 
tion of the force of gravity ; to be drawn 
at right-angles to the base b, and F par- 
allel to F. , 

By calculation see Formula, Fig. 181. 




185. 

w = the force with which the body 

presses against the plane, to be drawn 

at right-angles to the plane I ; then the 

parallelogram is completed. 

By calculation see Formula, Fig. 182. 



Wedge and Screws 



151 




186 



Wedge. 



F = 



Ra 



R = 



Fl 



I ' a 

F = force acquired to drive the wedge. 




187. 

Let the line F represent the magnitude and di- 
rection of a force acting to move the body B on the 
line CD; then the line a represents a part of F 
which presses the body B against CD, and the line 
b represents the magnitude of the force which 
actually moves the body B. 



b = ^F' — tf, b = Fcos.v. 




188. 

F : W = h : b = sin.r : cos.v : 

Wh 



= tan.v. 



F = 



= Wtan.v. F*-F. 



Fb F 

W= ~r = r— - F cot - v - 
h tan.-y 




189. Force by a Screw. 

P = Pitch of the screw, 
r = radius on which the force .Facts. 

Fi W=P:2/rr. 



F- WP 
*~ 27tr J 



W = 



F2nr 




190. Force hy Compound Screws. 

P = Pitch of the large screw, 
p = Pitch of the endless screw. 
R = radius of spur-wheel for the endless 
screw 

W: F^^Rr : Pp. 



F = 



WPi 



W= 



F^nRr 



4:n*R r 9 " Pp ' 

On the spur-wheel is a cylinder by which 
the weight W is wound up, the formula will 
be r' = radius of the cylinder, and 

•p :W=p r 1 ' :27t Rr. 
£ ~ 2* Rr 7 P r ' 



152 Dynamics. 

DYNAMICS. 

Velocity is a space passed through in a unit of time. 

In machinery velocities are measured in feet per second; for Steamboats and 
Railroads in miles per hour. 

Circular or angular velocity is the number of revolutions a revolving body 
makes per unit of time (minute.) 

Velocity of Men. 

A foot-soldier travels about 28 inches per step. 
In common time 90 steps per minute = 3*5 feet per second = 2-4 miles per hour. 
In quick time 110 " = 4*3 " =3 " 

Double quick time 140 " = 5*5 " = 3*75 " 

A soldier occupies in the rank, a front of 20 inches, and 13 inches deep with- 
out knapsack ; the interval between the ranks is 13 inches. 

Average weight of men, 150 pounds each. 

Five men can stand in a space of 1 square yard. 

Example 1, A man walked 450 feet in 75 seconds. With what velocity did he 
walk? 

s 450 
Formula 1. v = — = ■=— = 6 feet per second. 

t 75 

Example 2. A body moves 368 feet in t = 8 seconds. What velocity has it? 

„ , „ * 368 ,» * j. 

Formula 1. v = — = — = 46 feet per second. 

t o 

Example 3. The radius of a wheel is 4 feet and 4 inches ; it makes 131 revol- 
utions per minute. What is the velocity of a point in the circumference ? 

Formula 18. • = ^ = 2 X3-MX4-33 X 131 _ m feet secon(J> 

60 60 

Power is the product of force and velocity: that is, a force multiplied 
by the velocity with which it moves, is the power of the force. Force without 
velocity is no power. 

Force and Power are two distinct quantities. Power can not be increased or 
diminished by meehanical means; but Force can be increased and diminished 
ad libitum. 

When & force is in motion, and increased or diminished by mechanical means, 
the alteration will be at the expense of the velocity, so that the power will re- 
main the same. 

Horse Power* 

The unit for the measurement of mechanical Powers is assumed to be the 
power of a horse; or a force of 33000 pounds, moved through a space of one 
foot in one minute, or 550 pounds moved 1 foot in 1 second. Another unit for 
measuring small powers is, one pound moved through a space of one foot in one 
second ; this unit is called effect. ■ 

One horse power is = 550 effects. 

One man's power is = 50 effects. 

One horse power is = 11 men's power. 

Example 4. A man draws up a bucket of water which weighs 52 pounds, from 
the bottom of a well 83 feet deep, which space the bucket passes in 43 seconds. 
With what effect is that man working 1 

Formula 5. P = S = ^^~ = 100 Effects. 



Power, Force and Velocity. 



153 



Dynamical Formula* 



Velocity v = 

Space s < 

Time t = 

Power P = 

Power P = 

Force F = 

Force F = 

Velocity v -. 

Space s = 

Time t - 



s 

7' ' 


- 1, 


V t, - 


-2, 


s 
v 9 


- 3, 


vF, - 


-4, 


Fs 

t '" 


- 5, 


P 


•6, 


Pt 

s '" 


- 7, 


p 

F 9 ' 


-8, 


Pt 
F*~ 


■ 9, 


Fs 

T 9 " 


10, 



Fv 



Horse Power H= ^r-, 11, 

Fs 

Horse Power H= f^t, - 12, 



Force 

Force 

Velocity 

Space 

Time 



F-^-M, 13, 
F-***', 14, 



550 tf -, 

V = — y- , 15, 



550#* -„ 
s = — ^ — , 16, 



F 

Fs 

550iF 



17, 



Velocity t, -?*£»- 18, 

.Etfec* P= — gg — , 19, 

rr_F2 7trn _ JVn 
" 550x60 ~ 5250 ■ ^' 



Zefters denote. 



P = Power in j _ 

ZT^= Horse-power, number of. 

r = Velocity of the force Fin feet per second. 

t = Time in seconds. 

s = the space in feet which the force .F passes through in the time t. 

Circular Motion. 

r = radius of the circle in feet. 

n = number of revolutions per minute. 



l r 4 . Observed Results of Poweh. 




OBSERVED RESULTS OE 


POWER 


• 


hciirs 


Veociiy 


Effect*. 


HoTSiJi 




per 




V 


P 


H 


Description of works. 


day. 










A man can raise a weight by a single 












fixed pulley. 


6 


50 


0-8 


40 0-072 


A man working a crank, 


S 


20 


2-5 


50 


u-uyo 


A man on a tread-wheel, (horizontal. 1 ) 


8 


144 


0-5 


72 


0-130 


A man in a tread- wheel, (axis 24 D from 












vertical.) 


8 


30 


2-3 


— 
69 


0-125 


A man draws or pushes in a horizontal 












direction. 


S 


30 


2 


60 


0-109 


A man pull3 up or down. 


8 


12 


3-7 


44-4 


o-oso 


A man can bear on his back, 


7 


95 


25 


237-5 




A horse in a horse-mill, walking mode- 












rately, 


8 106 


3 


318 


0-577 


do'. do. do. running fast. 


5 72 


9 


64S 


0-165 


An ox do. do. walking mode- 










rately, 


8 154 


2 


308 


0-553 


A mule do. do. do. 


8 71 


3 


213 0-308 


An ass do. do. do. 


8 33 


2-65 


S7-4 0-160 


Flour Mills. 








For every 100 pounds of fine flour ground per hour, requires 


550 


1-000 


One pair of mill-stones of 4 feet diameter making l* 


20 rev. 






per minute, can grind 5 bushels of wheat to fine flour per hour. 


2400 


4-36 


Do. rye to coarse flour, 






2-91 


Saw Mills, alterative. 








For every 20 square feet sawed per hour, in dry oak 


there 






requires, 




550 


1-000 


Dry pine 30 square feet per hour. 




550 


1-000 


Circular Saw. 








A saw 2*5 feet in diameter ; and making 270 revolutions per 






minute will saw 40 square feet in oak per hour, with 




550 


1-000 


In dry spruce, 70 square feet per hour. 




550 


1-000 


Threshing Machine. 








Velocity of the feed rollers at the circumference 0* 


55 feet 






per second. Diameter of threshing-cylinder 3-5 feet 


and 4| 






feet Ions: making 300 revolutions per minute, can thres 


h from 






30 to 40 bushels of oats, and from 25 to 35 bushels of wheat. 






per hour. 




: 2200 


4-000 


One man by a flail can thresh half a bushel per hour, (wheat.) 


70 


0-127 


Rolling Mills. 








Bar iron-mills. Two pair of rough rollers, two pair of 


finish- 






ing rollers, six puddle furnaces, two welding furnaces, making 






lO^tons of bar iron per 24 hours, rollers making 70 revolutions 






per minute, requires 




29000 


80 


Plate-mill requires about five horses per square foot of 




1 


plates rolled. Largest size plate rollers should not 


make 


! 




over 30 revolutions^ er minute. 




i 1 



Dredging "Machines. 



155 



DKEDG-ING MACHINES. 



Letters denote. 

r=tons of materials excavated 
and raised per hour. 

h = night in feet in which the ma- 
terials are raised above the 
bottom of the excavated 
channel, 

k = 0*1 for hard clay with gravel. 

k = 0*07 for hard pure clay. 

k = # i;6 for common clay or sand. 

k — 0*04 for soft clay or loose sand. 

k = 0*03 for very loose materials. 

H= horse power required for ex- 
cavating and raising the ma- 
terials. 

F= force in pounds required to 
feet the Dredge ahead. 

v = velocity of the buckets in feet 
per second. 



formulas. 



H = r (w+*) 



T== 



700 g 
A+70J k 



F = 



F= 



550 H 



550 Tk 



T ' 700 



Example 1. What power is required to excavate T=160 tons of hard 
pure clay per hour, and raise it up h=25 feet above the bottom of the 
channel] For hard clay &=0*07. 

25 
H=160 (— -f-0-07) = 16-9, or 17 horses. 

Example 2. What force F=1 is required to feet the Dredge ahead tor 
the above example when the buckets move v=l foot per second. 
550X16-9 
1~ 



F=- 



• = 9295 pounds. 



LEATHEK BELTS. 

Letters denote, 
b = breadth of leather belt in inches, 
H== horse power transmitted by the belt. 
v = velocity of the belt in feet per second. 
d== diameter in inches ) __.- .. rt c , w „ 17rtC , + t,^ 1+ «„n rt „ 

n = revolutions per mintute } of the smaUest belt pulley. 
F= force in pounds transmitted by the belt. 
a = number of degrees occupied by the belt on the small pulley. 

7500 ff 
230 ' TaT * " 



H= V -L 
550 



H= 



dnF 

126500 ' 



F = 



12650012 
dn ' 



4 



13-5 v F 



b = 



d a 

n F 
1&8 ' 



6 = 



29nH 
v a 



8 



Example. Required the breadth of a leather belt to transmit a power 
of If -=75 horses over a pulley of 72 inches diameter, angle <t=170°. 

7400X75 4 r 4 . -u 

— — =4o*4 inches, the answer. 

72X1*70 5 



Fermulce 5. b = 



156 Collision of Bodies in Motion. 



COLLISION OF BODIES IN MOTION. 

When bodies in motten come in collision with each other, the sum of 
their concentrated momentum will be the same after the collision as 
before, but their velocities and sometimes their directions will differ. 

On the accompanying page the bodies are supposed to move in the 
same straight line, and the formula illustrates the consequences after 
collision. 

Letters denote. 

M and m = weight of the bodies in pounds. 

V and v = their respective velocities in feet per second. 

V and v f = respective velocties of the bodies after impact. 

K and k = coefficient of elasticity, which for perfectly hard bodies k^O 
and for perfect elastic bodies k=l, therefore the elastic coefficient will 
always be between and 1. When the bodies are perfectly hard their 
velocities after impact will be common. 

„ ... _ MV _ _ mv 

ForiT, K= — -— , For m, k = — — -. 

' M{V— F) ' m(v— V) 

Example 1. Fig 191. The non-elastic body weighs M=25 pounds, and 
moves at a velocity F=12 feet per second ; w=16 pounds, and v=9. Re- 
quired the bodies' common velocities, v'=1 after impact. 

MV±m^ 25X12+16X9 = ^ feet ^^ 
M-\-m 25-[-16 

Example 2. Fig. 195. The perfect elastic body 3f=84 pounds, F=18, 
m--=48, and u=27. Required the velocity F=l after impact with the 
body m. 

18 (84-48) -2X48X 27_ _ ^^ 

84+48 

the negative sign denotes that the body will return after the collision 
with a velocity of 2363 feet per second. 

Example 3. Fig. 196. The partly elastic body M==38 pounds and F=79 
feet per second, will strike the body in rest m=24 pounds ; what will be 
the velocity i/=l of the body m, its elasticity being &'=0'6. 

, 79X38(1+0-6) WA „_ . 
v'= — ^ — - = 70-6 feet per second. 

38+24 * 

When a moving body strikes a stationary elastic plane, its course of 
departure from the plane will be equal to its course of incident. 

^'•^ Vh Problem. A body in a is to strike the plane 

V* ,w u AB so that it will depart to the given point b ; 

x > x *>* \ required its course of incident from a? 

a vr^ !C- C Draw bd, at right angles through AB, make 

£ "^ i cd=bc join a and d; then ad is the course of in- 

v v, LV cident, and eb, the course of departure, and the 

* H * y body will strike in e. 



Impact of Bodies. Dynamics. 



157 



191. 

The bodies move in the same direction. 

v' (M+m) = MV+mv, 
V ~ ~M+m 




JL 



192. 

The bodies move in opposite directions. 

v* (M+m) = MV—mv, 
MV—mv 

V r =z . 

M+m 



193. 



V 



s~**&=0 



Only one body in motion, 
v* (M^m) = MV, 

MV 

v f = — . 

M+m 



194. 




The bodies move in the same direction. 
V(M—Km)+vm(l+K) 



F: 



M+m ' 

MV(1+Tc)+y (m—hM) 
M+m 



195. 




JV 



The bodies move in opposite directions. 
V(M—Km)—vm(l+K) 



P= 



M+m 
MV(l+h)—v (m—hM) 



M+m. 



V 



„-— N ir=o 



196. 

Only one body in motion. 

V(M—Km) 



V>= 



M+m 
VM(l+h) 

' M+m ' 



u 



168 Pile Driving. 



PILE DEIVING. 

Letters denote. 
M— weight of the ram in pounds. 
S =s fall of the ram in inches. 
m = weight of the pile in pounds. 
s = space in inches which the pile sinks by the blow. 
r — resistance of the ground in pounds. 
a = section area in sq. in. of the pile, sharpened to a point not more 

than 45°. 
k = coefficient for the hardness of the ground. 
h = depth to which the pile is driven. 
W— weight in pounds which a driven pile can bear with safety after 

the last blow when the pile sunk s inches. 
V= velocity in feet per second by which the ram strikes the pile. 
Ram and pilehead considered non-elastic and perfectly hard. 



V=8yS - - 1 

- 2 



' r (itf-fw) * 



W= * ps 



6s(Jf-j-m) 2 ' 



__ M*S 
~ s (M-j-m) 2 

r = ak y h 



Example 1. A wooden pile 18 feet long by 12 inches square, driven 
ft=12 feet into common natural ground imbedded with tenacious clay for 
which may be assumed the coefficient &=50. Required how much the 
pile will set s=1 into the ground at a blow with a ram of iH=3500 lbs. 
falling «S=42 inches. 

The weight of the wooden pile will be about m=18X40=720 lbs. 

Area of the pile a=l44 square inches. 

Resistance r=144X50/l2 =23840 lbs. 

The resistance sought from this formula 6, cannot be depended upon 
lor calculating the weight the pile can bear with safety. 

The set 5 = 350 ° 3X42 =4-23 inches. 

23840 (3500+720) 3 

Suppose the set to be s=4-23 inches at the last blow, required what 
weight the pile can bear with safety! 

3500 3 V42 

W'== ■ ^ „ = 3984 lbs. 

t>X4'23 (3500+720) Q 

This can be depended on with safety, if calculated from the actual set 
of the pile at the last blow. 

For ordinary pile driving a heavy ram and short fall is the most effec- 
tive, but in some cases when the ground itself is elastic, or when driving 
piles in pure sand it is found more advantageous to use a high fall of 
the ram. 

The Author has written to several places for information on pile driv- 
ing, in order to establish a table for the coefficient k; have received an- 
swers, but with no information. Hope to be able to give such a table in 
the next edition. 



Steam Hammer. 159 



STEAM HAMMER 

A heavy steam hammer with short fall produces a better forging than 
a light hammer with a high fall, although the dynamical momentum 
may be the.,same in both cases. This is accounted for by the inertia of 
the' ingot forged. 

The effect of blows of a heavy hammer and short fall will penetrate 
through the metal and nearly with the same effect on the anvil side, 
while a light hammer and high fall will effect the metal on or near the 
surface of the blow, because most of the momentum is in the latter case 
discharged in the inertia of the ingot forged. In forging large shaft it is 
generally piled up with iron bars sometimes rolled into a segment form 
to suit the pile, when placed under the hammer in a welding heat, very 
light and gentle blows are first given, then, the momentum of a light 
hammer may be discharged in the bars nearest to the blow, while a 
heavy hammer will squeeze the whole mass together throughout and a 
sound welding produced. 

The additional expense of a heavy hammer is fully compensated by 
the waste of labour and materials under a small one. I have often seen' 
in broken shafts the bars in the centre as clear and unwelded as when 
first piled, which is a pure indication that the shaft had been forged by 
a too light hammer. Crank shafts for propeller engines forged under a 
light hammer, when brought to the Machineshop the best part of the 
metal is worked away by plaining and turning, and the poorest left for 
the engine, but if forged under a heavy hammer, the difference of quality 
of metal will not be so great. 

Cases of this kind are well known m the U. S. Navy. 

Weight of Steam Hammers. 
The weight of a steam hammer in pounds should be at least 80 times 
the square of the diameter of the shaft in inches. 



CEMENT, CONCRETE AND MORTAR. 

Roman Cement. Parker's analysis. 

One part of common clay to 2£ parts of Chalk, set very quick. 

Concrete. Eight parts of pebble or pieces of brick about the size of 
an egg, to 4 parts of scrap river sand, and 1 part of good lime, mixed 
with water and grouted in, makes a good concrete. 

Lime Mortar. One part of river sand to two parts of powdered 
lime, mixed with fresh water. 

Hydralic Mortar. One part of pounded brick powder to two parts 
of powdered lime mixed with fresh water. This mortar must be laid 
very thick between the bricks, and the latter well soaked in water 
before laid. 

Hydralic Concrete, by Treussart. 

30 parts of hydraulic lime, measured in bulk before slacked. 

30 " Sand. 

20 " Gravel. 

40 _ " Broken stone, a hard lime stone. 

This concrete diminishes about one fifth in volume after manipulation. 

Asphalt Composition for street pavement by Colonel Emy. 

2£ pints (wine measure,) of pure mineral pitch. 

11 lbs of Gaugeac bitumen. 

17 pints of powdered stonedust, woodashes or minion. 



160 Friction. 

FRICTION. 

The resistance occasioned by Friction is independent of the Telocity of mo- 
tion ; but the re-effect of friction is proportional to the velocity. Friction is in- 
dependent of the extent of surface in contact when the pressure Remains the 
sarne, but proportional to the pressure. This law was established from experi- 
ments by Arthur Morin in the years 1831-32 and 1833, from which a summary 
is contained in the accompanying Table. 

Letters denote, 

a = Fibres of the woods are parallel to themselves, and to the direction of 
motion. 

b = Fibres at right-angles to fibres. 

c = Fibres vertical on the fibres which are parallel to the motion. 

d = Fibres parallel to themselves, but at right-angles to the motion, length 
by length. - 

e = Fibres vertical, end to end. 

Example. A vessel of 800 tons is to be hauled up an inclined plane, which 
inclines 9° 40' from the horizon ; the plane is of oak, and greased with tallow. 
What power is required to haul her up ? 

The coefficient for oak on oak with continued motion is / = 0*097, say 0*1, 
then, 

800Xsin.9° 40' = 800X0*16791 = 124*328 tons, 
the force required if there were no friction, and 

800Xcos.9° 40'XO'l = 80^<0-9858X0-l = 78*864 tons, 
the force required for the friction only, and 
134*328 
78*864 



213*192 tons, the force required to haul her up. 
on in axle and bearing 

p = 7td Wnf Wdn f 

12*60 230 > 



The effect lost by friction in axle and bearings is expressed simply by the 
formula 



in which W= the weight of pressure in the bearing, d = diameter on which 
the friction acts in inches, n = number of revolutions per minute, and/= co- 
efficient of friction from the Table. In common machinery kept in good order 
the coefficient of friction can be assumed to / = 0*065, then 

Wdn TT— Wdn — 

r ~~ 353 J ' J "~~ 1941500 

Example. The pressure on a steam-piston is 20000 pounds, and makes n = 40 
double strokes per minute. Bequired the friction in the shaft of d = 8 inchest 

„ 20000X8X40 _, ± ,. 

= — son " ~ 3*3 horses, the loss by friction. 

Friction in Guides. 

TT= pressure on the steam piston in pounds. 
S = stroke of piston in feet. 
I = length of connecting rod in feet. 
H= horse power of the friction. 

WSn 



H = . 



350000 v^p-^ 



Example. The pressure on a steam piston being W = 30,000 pounds stroke 
£=4 feet, length of connecting rod I = 7 feet, and making 50 revolutions per 
minute. Required the horse power of the friction II =1 

rr 30000X4X50 -, t1Q , 

=» 1 lo horses. 



"350000 y/ 5X1*^4* 



Friction. 



161 



TABLE OF FRICTION FOR PLANE SURFACES IN CONTACT. 


Kind of Materials in contact. 


Imbricated 
with. 


Goeffici 
Motion. 


cnt vi 
Starting. 


Oak on Oak, - 


a 





0-478 


0-625 


« " 








?? 


tallow 


0-097 


0-160 


a it , 








>> 


lard 


0-067 


.... 


tt tt 








b 





0-324 


0-540 


tt u 








}> 


unctuous 


0-143 


0-314 


tt tt 








» 


tallow 


0-083 


0-254 


tt a 










water, 


0-25 


.... 


tt tt 








d 





0'336 




tt tt 














0-192 


6-271 


tt tt 








e 







0-43 


C t-iron on Oak, 






a 





0-400 


0-570 


tt a 






jj 


soap 


0-214 


.... 


tt tt 






? ? 


tallow 


0-078 


0-108 


Wrought-iron on Oak, 






» 





0-252 


.... 


« tt 






>? 


tallow 


0-078 




Wrought iron, together. 






a 





0-138 


6-137 


it a 






a 


unctuous 


0-177 


. . . • 


it a 






» 


tallow 


0-082 


.... 


ti a 






t> 


olive oil 


0-070 


0-115 


"Wrought on cast-iron, 






a 





0-194 


0194 


a a 






» 


unctuous 


0-18 


0-118 


it il 






>> 


tallow 


0-103 


0-10 


ti it 






» 


olive oil 


0-066 


0100 


Cast-iron on cast-iron, 






a 


water 


0-314 


0-314 


a a 






n 


soap 


0-197 




tt ti 






:; 


tallow 


o-ioo 


6-100 


tt tt 






» 


olive oil 


0-064 


.... 


Wrought-iron on hrass, « 






a 





0-172 


.... 


a a 






n 


unctuous 


0-160 


.. .. 


tt tt 






jj 


tallow 


0-103 


.... 


a tt 






3J 


lard 


0-075 


.... 


a it 






,, 


olive oil, 


0-078 


.... 


Cast-iron on hrass, 






a 





0-147 


... 


a a . 






)5 


unctuous 


0-132 


.... 


a tt . 






5) 


tallow 


0-103 


.... 


tt tt . 






„ 


lard 


0-075 


. ... 


it u . 






,, 


olive oil 


0-078 


.... 


Brass on hrass, 






a 





0-201 


.... 


«• « - 






15 


unctuous 


0-134 


.... 


tt «... 






jj 


olive oil 


0-053 


.... 


Steel on cast-iron, - 






J? 




tallow 


0-202 
0-105 


.... 


tt a 






5? 


lard 


081 




it it 


a 


olive oil 


0-079 


- • 


FRICTION OF AXLES IN MOTION. 








OilyTc 


ittow, or Hog's Lard. 




Dry or si 


igMy 


Suppliec 


1 in the 


The grease 


Designation of surface in 


greasy, o 


rwet. 


ordin 


ary 


continually 


contacL 






mam 


%er. 


running. 


Brass on Brass, - 






0-07 


9 




" on cast-iron, - 






0-07 


2 


6*049 


Iron on Brass, - 




6»251 




0-07 


5 


0-054 


" on cast-iron, 








0-07 


5 


0-054 


Cast-iron on cast-iron, 




6*137 




0-07 


5 


0-054 


" on Brass, - 




0-194 




0-07 


5 


0-054 


Iron on lignum-vitse, 




0-188 




0-12 


5 




Cast-iron on " 




0.185 




o-ic 





6-092 


Lignum-vitas on cast-iron, 






Oil 


6 


0-170 



14* 



162 



Strength of Materials. 



STRENGTH OF MATERIALS. 

Table I., shows the weight a column can bear with safety ; when the weight 
presses through the length of the column. The tabular number is the weight 
in pounds or tons per square inch on the transverse section of a column cf 
a length less than 12 times its smallest thickness. 



Table I* 

RESISTANCE FOR COMPRESSION. 



197 



w\ 



Kind of Materials, 

Oak, of good quality, •.'•-- 

Oak, common, - 

Spruce, red (Sapin rouge), - 

" white, (Sapin blanc), 
Iron, wrought, - 
Iron, cast, - - - - 
Basalt, -•'•'.'...-•'- 
Granite, hard, - 

" common, - • • - - 
Marble, hard, ------ 

" common, - - - - - 

Sandstone, hard, - - • - • - 

" loose, - - - - - 

Brick, good quality, - . - - 

" common, - - - - - 

Lime-stone, of hardest kind, - - 

" common, - - - - 

Plaster-Paris, - - 

Mortar, good quality, and 18 months old, 
Do. common, - - - - - 

When the length or height of the column is more than 12 times its smallest 
thickness, divide the tabular weight by the corresponding number in this 
Table. 



Pounds. 


1 Tons. 


432 


0-1885 


280 


0-125 


540 


0-241 


140 


0-6256 


14400 


6-43 


28750 


12-85 


2875 


1-285 


1000 


0-446 


575 


0-256 


1435 


0-640 


431 


0-192 


1295 


0-577 


5-6 


0-0025 


175 


0-078 


58 


0*0259 


720 


0-321 


432 


0-193 


86 


0-0384 


58 


0-0259 


36 


0-016 



LengthXthickness 12 18 
Divide by V'Z 1-6 



Example. A building which is to weigh 2000 tons is to be supported by piles 
of Sapin rouge Spruce 18 feet in length, and 12 inches diameter. How many piles 
are required to support the building ? 



1 2^X0-785X0-241 
1-6 



= 17 tons, the weight which each pile can bear, 



&nd 



17 



= 118 piles. 



Professor Hodgkinson's Formulae for Crushing Strength of 
Cast Iron Pillars. 

The ends of the oillars should be perfectly flat and square, and the load to 
betir even on the whole surface. 
T=crushing weight in tous. 
JD=outside and d inside diameters in inches. 
/ =length or height of pillar in feet. 

!T=46-65 (— j^ ) 



Cast Iron Pillars. 



163 



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164 



Strength of Materials. 



Table II. 

COHESIVE STRENGTH PER SQ. INCH OE CROSS-SECTION. 





Just tear asunder. 


With sojhty. 




Kind of Materials, 


Pounds. 


Tons. 


Pounds. 


Tons, 


198 


Cast Steel, - 


13425b 
133152 


59-93 
59-43 


33600 
33300 


14-98 
14-86 


Blistered Steel, 


Y 


Steel, Shear, - 


128G32 


56-97 


32160 


14-24 






Iron, Swedish bar, - 


65000 


29-2 


16260 


7-3 






" Russian, 


59470 


26-7 


14900 


6-7 






" English, - ^ - 


56000 


25-0 


14000 


6-2o 






" common, over 2 in. sq., 


36000 


16-00 


9000 


4-0 






" sheet, parallel rolling, 


40000 


17-85 


10000 


4-46 






" at right angles to roll, 


34400 


15-35 


8600 


3-84 






Cast iron, good quality, - 


45000 


20-05 


11250 


5-00 






" inferior, - 


18000 


8-03 


4500 


2-0 






Copper, cast, - 


32500 


14-37 


8130 


3-6 






" rolled, 


61200 


27-2 


15300 


6-8 






Tin, cast, - 


5000 


2-23 


12500 


0-56 






Lead, cast, - 


880 


0-356 


220 


009 






" rolled, - 


3320 


1-48 


830 


0-37 






Platinum, wire, 


53000 


23-6 


13250 


5-9 






Brass, common. 


45000 


20-05 


11250 


5-0 






Wood. 














Ash, - 


16000 


7-14 


4000 


1-87 






Beach, - 


11500 


5-13 


2875 


1-28 


i 




Box, 


20000 


8-93 


5000 


2-23 


r^ 


Cedar, - 


11400 


5-09 


2850 


1-27 


/ \ 


Mahogany, - 


21000 


9-38 


5250 


2-34 


/w\ 


" Spanish, 


12000 


5-36 


3000 


1-34 


/ \ 


Oak, American white, - 


11500 


5-13 


2875 


1-28 


I 1 


" English " 


10000 


4-46 


2500 


1-11 




" seasoned, - 


13600 


6-07 


3400 


1-52 




Pine, pitch, . 


12000 


5-35 


3000 


1-34 




" Norway, - 


13000 


5-8 


3250 


1-45 




Walnut, - 


7800 


3-48 


1950 


0-87 




"Whalebone, - 


7600 


3-40 


1900 


0-85 




Hemp ropes, good, - 


6400 


2-86 


2130 


0-95 




Manilla ropes, 


3200 


1-43 


1100 


0-49 




Wire ropes, . 


38000 


17 


12600 


5-36 




Iron chain, - 


65000 


29 


21600 


9-38 




" with cross pieces, - 


90000 


40 


30000 


13-4 







To Find the Coliesivc Strength. 

Rule. — Multiply the cross-section of the materials in square inches by the 
tabular number in Table II., and the product is the cohesive strength. 

Example An iron-bar has a cross-section of 2-27 sq. in. How many tons are 
required to tear it asunder, and how many pounds can it bear with safety ? 

English iron 2*27X25 = 56*75 tons, which will tear it asunder, and it will bear 
with safety 

2-27X14000 = 31780 pounds. 







Chains, 


Heme 


AND 


Wire 


Ropes 






165 


Sufoty 


Inches and 16ths. 


Wht. 


per fathoin. 


Price per fathom, ultimate ' 


proof. 

C»vt. 


Chain. 


Hemp. ! Wire. 


Chain. 
Pounds 


Hemp. | Wire. 
Pounds Pounds 


Chain. 


Hemp. 


Wire. 


Strain. 
Cvrt. 


Diain. 


Circ'm. Circ.'m. 


$ cts. 


$ cts. 


$ cts. 


1.3 


1 


o-io 


0-4 


023 


0-08 


0-06 


0-15 


0-06 


0-08 


2-6 


4-5 


2 


1.6 


0-8 


0-93 


0-47 


0-24 


0-25 


0-12 


0*15 


9 


10 


3 


2-1 


0-12 


2-11 


1-06 


0-54 


0-36 


0-17 


0-22 


20 


18 


4 


2-12 


1-1 


3-75 


1-89 


1-10 


0-48 


025 


0-32 


35 


28 


5 


3-7 


1-6 


5-86 


2-94 


1-83 


0-60 


0-33 


0-43 


55 


40 


6 


4-2 


1-10 


8-45 


4-52 


2-56 


0-96 


0-42 


0-54 


80 


55 


7 


4-15 


1-14 


11:5 


6.09 


3-42 


1-25 


0-48 


0-62 


109 


69 


8 


5-8 


2-2 


15-0 


7-55 


4-39 


1-44 


0-60 


0-78 


138 


SO 


#9 


6-3 


2-6 


18-8 


9-56 


5-48 


1-80 


0-76 


0-90 


160 


94 


10 


6-14 


2-11 


23-0 


11-8 


7-00 


1-86 


0-95 


1-20 


218 


109 


11 


7-9 


2-15 


27-7 


14-3 


8-38 


2-16 


1-14 


1-50 


187 


127 


12 


8-4 


3-3 


33-0 


171 


9.90 


2-43 


1-37 


1-80 


254 


147 


13 


8-15 


3-8 


38-5 


199 


11-9 


2-70 


1-60 


2-10 


293 


168 


14 


9-10 


3-12 


44-7 


23-1 


13-6 


3-06 


1-85 


2-28 


335 


199 


15 


10-5 


4-1 


51-1 


26-3 


16-0 


3-70 


2-10 


2-45 


397 


220 


1 * 


11- 


4-6 


58-0 


30-2 


18-6 


4-33 


2-42 


2-73 


440 


246 


1-1 


11-11 


4-11 


65-6 


34-1 


21-3 


4-68 


2-73 


310 


492 


278 


1-2 


12-6 


5 in. 


73-7 


38-2 


24-2 


5-58 


3-06 


3-50 


545 


302 


1-3 


13-1 


5-5 


82-1 


42-6 


27-4 


5-86 


3-40 


3-91 


604 


332 


1-4 


13-12 


5-10 


91-0 


47-1 


30-7 


6-42 


3-77 


4-35 


663 


365 


1-5 


14-7 


6 in 


100 


52-0 


35- 


7.08 


4-16 


4-89 


730 


399 


1-6 


15-2 


6-5 


110 


57-1 


38-7 


7-75 


4-57 


5-35 


798 


435 


1-7 


15-15 


6-10 


120 


63-4 


42-6 


8-42 


5-07 


5-83 


869 


472 


1-8 


16-8 


6-15 


131 


67-9 


46-7 


915 


5-44 


6-35 


944 


553 


1-10 


17-14 


7-10 


154 


79-8 


56-4 


10-07 


6-38 


763 


1105 


638 


1-12 


19-4 


8-4 


178 


92-6 


66-0 


12-38 


7-40 


8-83 


1275 


729 


1-14 


20-10 


8-14 


205 


103 


76-5 


14-15 


8-48 


10-00 


1457 


825 


2 in 


22- 


9.8 


232 


121 


88-0 


16-00 


9-70 


11-50 


1650 


1072 


2-4 


24-12 


10-12 


293 


153 


112 


20-75 


10-25 


14-60 


2141 


1288 


2-8 


27-8 


12 in. 


363 


189 


140 


25- 


15-10 


18-00 


2575 


1559 


2-12 


30-4 


13 4 


438 


229 


172 


30-25 


18-30 


21-80 


3117 


1854 


3 in |33« 


14-8 


522 


272 


205 


36-00 ; 21-80 


25-90 


3708 


The prices of the chains are ta' 


ten frc 


m thai 


in England and added 50 


per cent. Price of hemp ropes f 


rom \^ 


r eaver, 


Fitler & Co., Rope manu- 


facturers, Philadelphia. The p 


dees o 


f Wire 


i ropes are deduced from 


the price list of John A. Roebl 


ing, P 


atent 


Wire Rope Manufacturer, 


Trenton, N. J. 








The Safety proof is here taken 


one ha 


lfofth 


e ultimate strength which 


may be trusted on for new ropes, 


but w 


hen mi 


ich in use only one quarter 


or less 


shoul 


1 be tn 


isted u 


pon fo 


r safet 


y- 




*> 







166 




Cables and Anchors. 






CABLES AND ANCHORS. 


Table showing tlie size of Cables and Anchors proportioned to the Tonnage of 


Vessels. 


Tonnage oj 
Vessels. 


Cables. 

Circumference 

in inches. 


Chain Cables 

Diameter in 

inches. 


Proof 

in 
tons. 


Weight of 
Anchor in 
pounds. 


Weight of 

fathom of 

Chain. 


Weight of a 

fathom of 

Cables. 


5 


3- 


. 5 
IB" 

. 3 
8 

•t 7 5 


.3 


56 


5f 


2-1 


8 


4- 


If 


84 


8- 


4- 


10 


4'i 


*ft 


112 


11- 


4-6 
6-5 


15 


5-i 


4- 


168 


14- 


25 


6- 


. 9 


5. 


224 


17- 


8-4 


40 


6* 


. 5 

"8" 


6- 


336 


24- 


9-8 


60 


7- 


.11 
1 S 


7- 


392 


27- 


11-4 


75 


7-J 


. 3 


9- 


532 


30- 


13- 


100 


8- 


.13 
T6 


10- 


616 


36- 


15* 


130 


9- 


. 7 

8 

.15 

1U 


12- 


700 


42- 


18-9 


150 


n 


14- 


840 


50- 


21- 


130 


ttf 


V 


16- 


952 


56- 


25*7 


200 


li- 


^ 


18- 


1176 


60- 


28-2 


210 


ra 


**l 


20- 


1400 


70- 


33-6 


270 


in 


X 'A 


21- 


1456 


78- 


36-4 


320 


13$ 


I'* 


22-* 


1680 


86- 


42-5 


360 


14- 


**A 


25- 


1904 


96- 


45-7 


400 


m 


!•# 


27- 


2072 


104- 


49- 


440 


15*i 


»'A 


30- 


2240 


115- 


56- 


480 


16- 


I"* 


33- 


2408 


125- 


59-5 


520 


i6-a 


i-A 


36- 


2800 


136- 


63-4 


570 


17- 


T 5 

1 S" 


39- 


3360 


144- 


67-2 


620 


17'i 


i"i§ 


42- 


3920 


152- 


711 


680 


18- 


i-i 


45- 


4200 


161. 


75-6 


740 


19- 


*•** 


49- 


4180 


172' 


84-2 


820 


20- 


i-i 


52- 


5600 


184- 


93-3 


900 


22- 


i-il 


56- 


6720 


196- 


112-9 


1000 


24- 


2- 


60- 


7168 


208- 


134-6 


The proof in the U. S. Naval Service is about 12^ per cent, less than the above 
for the larger sizes, and from 25 to 30 per cent, for the smaller. 


The results of experiments at the U. S. Navy Yard, Washington, D. C, give 
for the cohesive force of chain iron, per square inch, as follows : 


Mean of experiments with good iro 
Mean of experiments with best iron 


Q ...... .....^ 


1000 lbs. 
3000 lbs. 


/ 4 


- - . J 





Weight and 


Shrinking or Castings. 167 


To find tlie weight of Castings, by the weight of 


Pine Patterns. 


RULE. 


{ 12 for Cast Iron, \ 


Multiply the weight \ 13 « Brass, / and the product ig the 


of the Pattern by J 12-2 « Tin, ' I weight of the Castings. 


V 11-4 " Zinc, / 


Reductions for Bound Cores and Core-prints. 


Rule. Multiply the square of the diameter by the length of the Core in 


inches, and the product by 0-017, is the weight of the pine core, to be deduc- 


ted from the weight of the pattern. 


Shrinking of Castings* 


Cast Iron, £ ^ 


I Brass 3 ^ 
Pattern Makers' Rule \ Lead ' \ 5 .S 1 longer per Linear 


should be for # Tin ' J 8 ( Foot. 


' Zinc, T % ) 


Weight and Capacity of Balls* 


Diameter in 


Capacity in cubic 


CAST IRON. 


LEAD. 


inches. 


inches. 


Pounds. 


Pound s. 


1- 


•5235 


•1365 


•2147 


l'i 


1-7671 


•4607 


•7248 


2- 


4-1887 


1-0920 


1-7180 


2-i 


8-1812 


2-1328 


3-3554 


3- 


14-1371 


3-6855 


5-7982 


3* 


22-4492 


5-8525 


9-2073 


4- 


33-5103 


8-7361 


13-744 


4'i 


47-7129 


12-4387 


19-569 


5* 


65-4498 


17-0628 


26-843 


5*' 


87-1137 


22-7206 


35-729 


6- 


113-0973 


29-4845 


46-385 


6'i 


143-7932 


37-4528 


58-976 


7- 


179-5943 


46 8203 


73-659 


n 


220-8932 


57-5870 


90-598 


8- 


268-0825 


69-8892 


109-952 


n 


321-5550 


83-8396 


131-383 


9- 


381-7034 


99-5103 


156-553 


h 


448-9204 


117-0338 


184-121 


lO- 


523-5987 


136-5025 


214-749 


ll- 


696-9098 


181-7648 


285-832 


12- 


904-7784 


235-8763 


371-096 


13- 


1150-346 


299-6230 


471-806 


14- 


1436-754 


374-5629 


589-273 


15- 


1767-145 


460-6959 


724-781 


16- 


2144-660 


559-1142 


879-616 


17- 


2572-4 '0 


670-7168 


1055-066 


18- 


3053-627 


796-0825 


1252-422 


19- 


3591-3d3 


936-2703 


1472-970 


20- 


418S-790 


1092-0200 


1717-995 



183 



WftOUGHT IllON BEAMS. 



LATEEAL STRENGTH. 

For wrought iron beams, letters denote. 

Wz= weight in tons with safety, uniformly distributed on a beam rest- 
ing on two supports. 

S = compressive strain in tons per square inch of top 0*5 (a-\ — .\ 

a = section area in square inches of top and bottom flanges' of the 

beam. Top and bottom flanges to be alike. 
a'— section area in square inches of web or stem. 
h = height of beam in inches. 
I = length in feet between supports. 
/= deflection in inches of the beam in the centre, when the weight is 

uniformly distributed. 
P = weight in pound per square foot of flooring to be supported by the 

beams, which in ordinary cases is estimated to P=140 lbs. 
B = distance in feet between the beams. 
w = weight of the whole beam in pounds. 



W- 



■-■r (•**)'- 



T __ STi r , a' N 



s= 



ZW l 



f= 



f= 



WP 



46 /t 2 (3 a-\-a') ' 



46ft* 



2240 W 



to = 3-384 l(a-\-a'.) 



Formula 6 gives the safety deflection of a wrought iron beam, which 
should never be exceeded. 

Example. A flooring of Z=32 feet by 60 feet long to be constructed to 
support P=175 pounds per square foot. Required what kind of beams 
and how many are necessary] and what will be the cost of them] 

In the table will be found the nearest star to 32 feet span is a 12 inch 
beam bearing W=8'71 tons, when the distance between the beams in 
the flooring will be, 



Formula 7. B = 



2240X8*71 
175X32" 



= 3*5 feet. 



Number of beams = ■ — 1=16 about. 

3-5 
Add one foot to each beam for the supports at the ends, and the cost 
will be, 33X16X1-90=1003-70 dollars. 



The following Table contains sections of iron rolled by the Phoenix 
Iron Company. Orflce 410 Walnut Street, Philadelphia. 

Rules. 

The price per foot multiplied by 5280 gives the price per mile. 

The weight in pounds per foot multiplied by 2-36 gives the weight in 
tons per mile. 

The price per foot multiplied by 2240 and divided by the weight in 
pounds per foot gives the price per ton. 



Strength of Different Sections of Wrought Iron Beams. 



169 



Strength of different Sections of Wrought Iron Beams 


Made by the Phoenix Iron Company, for Sustaining with Safety 




a Load Uniformly Distributed. 






Compound Girders. 


Solid Rolled Beams. 




Dis. 


800 667 553 


490 


296 308 168 84 


TTT 48 


bet! 


,r=- r 


TT=- r 


W -T 


W=-j- 


w= T 


w=- r 


*=T 


T 


W =T 


sup 
=1. 


ft=18 i. 


fc=15 i. 


%==12 i 


/i=15 i. 


A=12 i. 


h=9 i. 


h=9 i. 


h=i i. 


h = % i. 


feet. 


tons, 1 tons. 


tonB. 


tons. 


tons. 


tons. 1 tons, 


ton*. 


tons. 


10 


80-00 ! 66*67 


5533 


49-00 


29-60 


30-80 16-80 


8-40 


4-S0 


12 


66-66 


55-56 


44-44 


40-83 


24-66 


25-69 14-00 


7'00 


4-00 


14 


57-14 


47*61 


38-09 


35-00 


21-14 


22-05 


12-00 


6-00 


3-43* 


16 


50-00 


41-67 


33-33 


30-63 


18-50 


19-25 


10-50 


5-25 


300 


18 


44-44 


37*04 


28-52 


27-22 


16-44 


17-11 


9-33 


4-66 


2-66 


20 


40-00 


33-33 


26-66 


24-50 


14-80 


15-40 


8-4 


4-20 


2-40 


22 


3636 30-30 


2424 


22-27 


13-45 


14-00 


7-63 


3-81* 


2-18 


24 


33-33 


27-77 


22-22 


20-42 


12-33 


12-85 


7-00 


3-50 


2-00 


26 


3^77 


25-64 


20-05 


18-85 


11-38 


11-87 


6-46 


3-23 


1-84 


28 


2^7 


23-80 


19-05 


17-50 


10-57 


11-00 


6-00* 


3-00 


1-71 


30 


26-66 


22-22 


17-77 


16-33 


9-86 


10-26 


5-60 


2-80 


1-60 


32 


25-00 


20-83 


16-66 


15-31 


9-25 


9-62 


5-25 


2-62 


1-50 


34 


23-53 


19-60 


15-65 


14-41 


8-71* 


9-06* 


4-94 


2-47 


1-40 


36 


22-22 


18-52 


14/26 


1361 


8-22 


8-55 


466 


2-33 


1-33 


38 


21-05 


17-37 


14-00 


12-90* 


7-80 


8-11 


4-42 


2-21 


1-26 


40 


20-00 


16-66 


13-33 


12-25 


7-40 


7-70 


4-20 


2-10 


1-20 


42 


19-05 


15-87 


12-70* 


11-67 


7-05 


7-34 


4-00 


2-00 


114 


44 


18-18 


15-15 


12-12 


11-13 


6-72 


7-00 


3-81 


1-91 


1-09 


46 


17-37 


14-48* 


11-44 


10-66 


6-43 


6-70 


3-65 


1-83 


1-04 


48 


16-66 


13-88 


11-11 


10-21 


6-16 


6-42 


3-50 


1-75 


1-00 


50 


, 16-00* 13-33 10-66 


9-80 


592 ' 6-16 


3-36 


1-68 


•96 


Per 


78 lbs. 1 71 lbs. 59 5 lbs 


47-8 lbs. 40 lbs. 60 lbs. 29 3 lbs. 20 lbs. 


13 3 lbs. 


Foot' 


$4'68 $4-26 $3-57 


j $2-40 ; $1-90 $2-40 $1'50 ; $1'00 


$0.66 


The above Table gives the weight in tons, sustained by the 


several 


kinds of beams, uniformly distributed over them as in a floor 


. The 


weights given are what may be used in practice, being only 9 t 


ons per 


square inch of that part of the metal subjected to a crushing fore 




Under these weights the beams are within the limits of perfe( 


;t elas- 


ticity, and the deflections are therefore in direct proportion to the 


i load. 


If it be intended to apply the entire weight at the centre, the 


figures 


in the Table must be divided by two ; if at any other point, the 


weight 


at the centre is to the weight at any other point, as the square 


of half 


the beam is to the rectangle of the two parts from where the we 


ight is 


applied. The prices are subject to changes of the market and 


agree- 


ment. 




* When the span of the flooring is given, the star in the Tab' 


e gives 


an approximation to what beam ought to be employed; for in 


stance, 


J=38 feet span should have beams of fc=15 inches high, able 


to bear 


W=-- 12*9 tons uniformly distributed. 





15 



170 



Dimensions, Weight, and Price of Rolled Iron. 



Angle Iron. ! 


Variety of Forms. | Price 


Per Foot- 
Dimensions. Weight. Price. 


Section. 


Area. iWtp.ft. 


Wt.p.Mile. 


Per Ft. 


Inches. 


lbs. 


cents. 




Figure, 


Sq, In, 


lbs, 


Tons 


$ eta, 


ft('*+U) 


1-77 


5-13 


1 


7-4 


25 


11786 


0-56 


i (U+U) 


2-32 


6-74 


2 


XL 


5-9 


20 


92-7 


0-45 


^(li+ll) 
^(li+li, 

1 ( 2+2 ) 
f (2+2) 

T S g( 2i+2i ) 


2-09 
3-49 
3-17 
4-59 

4.97 


6-07 
10-1 
8-60 
13-3 
14-4 


3 

4 


JL 

2. 


7-1 

« 

1-95 


24 
6-6 
18-3 
15 


113.14 
31*45 

86-43 
70-71 


0-54 
0-16 
0-45 
0-37 


5 

6 


J^^ mmmm *\ O rtL 


T T g (24+2J) 
|(3+3) 

H 3+3 ) 

3 (3i+34) 


6.84 
7-13 
9-32 
8-40 


19-9 
20-7 
27-1 
24-4 


7 

8 

1 

9 

i 


^ 


4-22 
7-00 
5-32 


14-3 
23-6 

18 


67-75 
111-5% 
Chair. 


0-35 
0-58 
0-72 


LJ 


r V(3i+34) 


12-2 


34-9 


10 


9-65 


32.6 


Channel. 


1-16 


r 7 s( 4+<t ) 


11-2 


32-5 


1 
11 


L-J 


5*41 


1S-3 


Channel. 


0-65 


1(4+4) 


15-5 


45-0 


12 


X 


2-66 


9 


Purlin. 


0-35 


ShipFrames. 






IS 
11 
If 


± 


2-66 


9 


T iron. 


0-32 


|-2i+ T 5 g -31 


2.5 

4-36 
6.68 


7-95 
13-8 
21-2 


a W 3sam 


0-65 
0-50 


2-2 
1-7 


Window- 
Sashes. 


12 
12 


fU+A^ 


8.85 


28-1 


u 


H- 


0-89 


3 


Sash bar. 


12 


/,••+!•« 


11-0 


35-0 


Vi 




2-07 7 


Shoe. 


0-25 






P+iV 6 


16-4 


51-0 


16 


i ,*^P^- 


6-66 J 22-5 


Girder. 


0-80 


) 4UcM 

) 


■■•'A 
h 


This is the beam for which 
the formulas and table are set 
up. Top and bottom are alike. 








21 


1 

C 
I 


r ?1 




This compound Girder is 
made to order of any size,. for 
about 6 cents per pound. 










JLv 


# 20 intermediate siaes, 



Strength of Materials. 171 



LATERAL STRENGTH OF MATERIALS. 

The formulas for lateral strength are here reduced to the simplest pos- 
sible form, and are in consequence subject to conditions which must be 
particularly attended to. In calculating the strength of beams of ir- 
regular sections as shown by the figures 210 to 217 on page 173, it is neces- 
sary to maintain the proportions marked on the figures and the calcu- 
lation will be correct. For the sections 206 to 209 any proportion will 
answer in the formulas. The weight of the beam itself has not here 
been taken id to consideration, for which allowance must be made if 
considerable. 

Letters denote. 

I = length of beam in feet. See figures. 

h = height, &=breadth or thickness in inches of the beam, where the 

strain is acting. 
k = coefficient for the different materials and sections of beams, to be 

found in the tables. 
x — modulus of elasticity of materials. See Table. 
/= elastic deflection in inches. 

W= weight in pounds which the beam can bear with safety, being 

about one quarter of the ultimate strain at which the beam 

would break. 

Example 1. Fig. 200. A rectangular beam of oak fastened in a wall 

projects outZ=6 feet 4 inches, fi=8 inches, and 6=5 inches. Required 

what weight it can bear on the end W=1 

W= i°X°X81 = 1509 pounds wit h perfect safety. 
6-333 ' 

Example 2. Fig. 201. A beam of section fig. 211, with thickness 6=1-25 
inches, height ft =22-5 inches, supported at the two ends in a length 1=25 
feet. Required what weight W^ it can bear in the middle. For cast 
iron coefficient £=260. 

^4X260X1-25X22-5^ ^ ^^ ^ ^ 

Example 3. Required the elastic reflection for the same beam and con- 
dition as in the foregoing example ] See Table, modulus of elasticity 
#=2285 for cast iron. See page 176. 

/= 26325X25^ = QtQQ ^ nea rly. 

16X2285X1-25X22-5 3 

Example 4. Fig. 204. A wrought iron girder of section fig. 217, consist- 
ing of four angle irons of &=3 , 5X0 , 5X2X4=14 square inches, the plate 
being 0'5:1*35=0'37 inches thick, and ft=18 inches deep by Z=22 feet. Re- 
quired how much weight evenly distributed the girder can bear with 
safety ? 

8X8QQXUX18 =73309 lbSi=33 . 75 tons . 

22 

If plates being riveted to the angle iron at top and bottom, add that 
area to a. 

Example 5. Fig. 222. The crank E=3'5 feet, force F=3860 lbs., length 
of the shaft Z=64 feet, diameter D=5-25 inches. Required the twisting 
in degrees. The shaft being of wrought iron for which #=4110. 

Degrees = i^ 6 ° X3 ' 5 ^ = llW. 
4110X5-25* 



172 



Strength and Elasticity of Materials. 






c 




3? 



_E 



¥ 



tzt 



tyu ^ 






s 



in 



200. 



it-*** i*@t+o 



wr 

f= — 7-Ti-- elastic set in inches. 



#6 A 3 



201. 






/= 



I 

TFZ* 



4 wi i 



"16 a: 6 A 3 



elastic set in inches. 



HI 




202. 



TF= 



l **"br / 



"cr 



fe 



8 k 17? 

~T~' 

wr 



W>~- 



w? 



4 Wl 



7- T 



'32 a: 6 fr 



. elastic set in inches. 




203. 



W= 



2 h b A 2 



I 



I#®T+D 



, 5 T7 Z 3 , ■ . . . , 

f=.— — — — ; elastic set in inches. 



2 x b A 3 



204. 



17= 



Skbh* 



I 



1 



5 TF Z 3 
f=7^ — r-7-3- elastic set m inches. 



~Z2xbh'' 



ti:_ 205. 

Ec w= 



16 I- & A 9 



I 



T 



„ 5 TTZ 3 

f=-^- A — r-TT . elastic set m inches. 
J 64 ar b A 3 



Different Forms of Beams.. 



173 




206c 

Coefficient Jc. 

Cast iron, 150 
Wro'tiron, 120 
Wood, 30 




212. 

Coefficient Jc. 

Cast iron, 236 
Wro'tiron, 189 




207. 

Coefficient Jc. 
Cast iron, 150 
Wro'tiron, 120 
Wood, 30 

b A— & 3 . 




213. 

Coefficient Jc. 

Cast iron, 250 
Wro'tiron, 200 




208. 

Coefficient Jc. 

Cast iron, 88 

Wro't iron, 70 

Wood, 18 

b A °=Z> 3 . 




12 b A 



214. 

Coefficient Jc. 

Cast iron, 700 
Wro't iron, 560 




209. 

Coefficient Jc. 

Cast iron, 88 

Wro't iron, 70 

Wood, 18 

b A 3 =£ 3 — ef. 




'215. 
Coefficient Jc. 

Cast iron, 900 




210. 

Coefficient Jc. 

Wro'tiron, 700 

b #=a h. 



^<6l 




211. 
Coefficient Jc. 



Cast iron, 260 
Wro'tiron, 208 




216. 

Cast iron tube, 

£=800. 




7.35 



217. 

£=800, 

b ff=a hj 

ac=area of all 
the four angle 
irons in square 
inches. 



15* 



174 



Strength of Materials. 




218. 

A beam fixed in one end and loaded 
at the other, should have the form of a 
Parabola, in which Z = abscissa and k== 
ordinate. y= depth, x= length from W, 



'- 1 */*- 




219. 



W= 



I cos.v 



h b h* 

= b' ' 




220. 



TF= 



36 h b A» 



I 



Divide the length into 24 equal parts, 
place 14 in the middle and 5 at each end. 




221. To cut out the stoutest rectangular 
beam from a log. 

1st case, divide the diameter in 3 equal 
parts, and draw lines at right-angles as 
represented. 

2d, divide the diameter in 4 equal parts. 

1, 5=1-414 6, non-elastic. 

2, /z=l*73 6, elastic beams. 



222. 




D=4a 



Twisting in degrees = 



425 FR I 



xD* 




223. 

3 f~H 

D=$Q\ / , 

y x n 

Twisting in degrees : 



2 233000 E I 



Diameter, of VTftocanT Titox Shafts in Inches, 



175 





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176 



Strength op Materials. 



Absolute and Ultimate Strength of Materials. 



Kind of Materials. 



\ Safety. 



Wrought iron - 
Cast iron, 
Cast steel, soft, - 
Cast steel, hardened. 
Blasted steel, soft, 
Brass, - 
Copper, 
Zinc, - 

Tin, - - 
Lead, - 
Ash, - 
Hickory, - 
Chestnut, sweet, 
Oak, white, 
Oak, English, 
Canadian Oak, 
Pine, white, 
Yellow, pine, - 
Teak, - 



120 

150 

385 

1050 

175 

58 

53 

15 

17 

4 

45 

67 

42 

50 

25 

37 

34 

38 

51 



Coefficient k, 
Inter. \Pr. cir. [Ultimate 



-i- 



162 

200 

619 

1400 

233 

75 

71 

20 

23 

6 

56 

90 

56 

66 

33 

49 

45 

50 



240 

300 

170 

2100 

350 

113 

106 

30 

34 

9 

85 

135 

85 

100 

50 

73 

67 

75 

102 



600 

1540 

4200 

700 

226 

212 

61 

69 

18 

170 

270 

170 

200 

100 

147 

135 

150 

205 



Elasticity. 



w 



4110 
22S5 
4300 
6000 
4200 
1280 
2160 
2360 

100 
221 



300 
248 
283 

268 
316 



The absolute safety weight is here taken one quarter of the ultimate 
breaking weight, but when the weight is acting at short intervals one 
third might be relied upon, or in pressing circumstances one half, when 
the materials in the beams are known to be of good quality ; but the 
latter never to be exceeded. 



BRICKS. 

Dimensions. 
Common brick 8X4JX24 inches — 85 cubic inches. 
Front brick 8iX4£X2£ „ = 92*8 „ „ 
When laid in a wall with cement it occupies a space of 
Common brick 84X4^X21 inches = 102 cubic inches. 
Front brick 8£X4|X2i „ = 111 „ „ 
Weight and Bulk of Bricks. 









Number of bricks, 


Tons. 


Pounds. 


Cub. feet. 


by itself. 


in wall wi 


th cement 








C. Brick. F, Brick. 


C. Brick. 
381 


F. Brick. 


1 


2240 


22-4 


448 


416-6 


347 


0-04464 


100 


1 


20 


18-6 


17 


154 


2-23 


5000 


50-00 


1000 


930 


850 


772 


2-4 


5376 


53-76 


1075 


lOOO 


914 


834 


2-62 


5872 


58-72 


1130 


1100 


1000 


913 


2*88 


6451 


64-51 


1240 


1200 


1100 


1000 



A Cord of wood is 4 feet wide, 4 feet high, and 8 feet deep, 
128 cubic feet. 



Steam Boilers. 



177 



Inspector's Table for Steam-Boilers on Western River*. 



Tliickness 

iron. 

W, g, inches, 



No, 
1 

2 
3 

4 
5 
6 

7 



in, 

0-300 
0-284 
0-259 
0-238 
0-220 
0-202 
0-180 



Diameter of boiler in inches. 



34 


36 


38 


40 


42 
~~ lbs, - ~ 


44 | 


lbs, 


lbs, 


lbs, 


lbs, 


lbs. 


169-8 


160-4 


151-9 


144-3 


137-5 


131-2 


158-5 


149-7 


141-8 


134-7 


128-3 


122-5 


147-2 


139-0 


131-7 


125-1 


119-1 


113-7 


138*8 


128-3 


121-5 


115-5 


110-0 


105-0 


124-5 


117-6 


111-4 


105-8 


100-8 


96-2 * 


113-2 


606-9 


101-3 


96-2 


91-6 


87-5 


101-9 


96-2 


91-1 


86-6 


82-5 


78-7 



46 

~lbs^ 

125-5 

1171 

108-8 

100*4 

92-0 

83-6 

75-3 



Working steam pressure per square inch. 



The following table is given by Mr. Fairbairn, as exhibiting the 
strongest form and best proportions of Riveted joints, as deduced from 
experiments and actual practice. 



Thickness 


Diameter of .Length of rivet Distance from 


Quantity of lap in 


of plate. 


rivet. 


fro. head, 


centre to cent. 


single riveted. 


double riveted. 


in, 16ths, 


in. 


Ratio, 


in. 


Ratio. 


in. 


Ratio. 


in. 


Ratio. 


in. 


Ratio, 


0-19= 3 


0-38 


2 


0-88 


4-5 


1-25 


6 


1-25 


6 


2.10 


10 


25= 4 


0-50 


2 


1-13 


4-5 


1-50 


6 


1-50 


6 


2-50 


10 


0-31= 5 


0-63 


2 


1-38 


4-5 


1-63 


5 


1-88 


6 


3-15 


10 


0-38= 6 


0-75 


2 


1-63 


4-5 


1-75 


5 


2-00 


5-5 


3-33 


9-2 


0-50= 8 


0-81 


1-5 


2-25 


4-5 


2-00 


4 


2-25 


4-5 


3-75 


7-5 


0-63=10 


94 


1-5 


2-75 


4-5 


2-50 


4 


2-75 


4-5 


4 58 


7-5 


0-75=12 


113 


1-5 


3-25 


4-5 


3-00 


4 


3-25 


4-5 


5-42 


7-5 



178 



Gearing. 



GEARING. 




Letters denote, 
P = pitch, —the distances between the centres of two teeth In the 

pitch circle. 
D = diameter 
C = circumference 
M== number of teeth 
iV= number of revolutions 
d = diameter 
c = circumference 
m s= number of teeth 
n — number of revolutions 



of the wheel. 



> of the pinion. 



Pitch 



No. of teeth ■ 



P = 



a 

M 

it D 

C_ 
P 

nD 



Circum, 



Diameter 



(C=PM 



D = 



PM 






D:d= C:c = M:m = n:N 

Example 1. A wheel of D = 40 inches in diameter, is to have M = 
75 teeth. Required the pitch JP= ? 

3-14X40 



Formula 2. Pitch P= 



75 



-= 1*66 inches nearly. 



Example 2. The pitch of teeth in a wheel, is to be P = 1*71 inches, and 
having if = 48 teeth. Required the diameter D = ? of the wheel. 
■ 1 •'71X4-8 

Formula*!. Diam. D — — =26'14 in. of the pitch circle 

O* J.4: 



Gearing. 179 



Construction of Teeth for Wheels. 

Draw the radius R r, and pitch circle P P. Through r draw the line o o' at 
an angle of 75° to the radius E r. 

, ,,„ 180 , 

e wheel, v = — =y- . - - - 1 

Half the angle be- 1 M 

tween two teeth in the 1 jgQ 

Opinion, F=-^— .--. 2 

D : d = sin, V : sin, v, 

r wheel, I? = — : . - - 3 

I sin, v 

Diameter of the < 

I . . . D sin, v 

Opinion, d = ^ y • -.. 4 

Pitch (chord) of teeth f wheel, P = D sin, v, - - - 5 
in the pitch circle I pinion, P= c? m F. • • -6 

Approximate pitch in the wheel P = 0*028 Z). - - • 7 
r wheel, Jlf-?^?.. ... 8 



j 



Number of teeth about 

I . . dM 

Opinion, m = - j, ■•- • • - 9 

Thickness of tooth, a = 0*46 P* - 10 

Bottom clearance, h = 0'4 P. 11 

Depth to pitch line, c = 0*3 P. 12 

Distances, ^=^^ 13 

Distance ro', e = 0*11 P fym 14 



* If a wheel of more than 80 teeth is to gear a pinion of less than 20 teeth, 
and the wheel and pinion are of the same kind of materials ; take the thickness 

r wheel, a = P (o*42 + ~-\ . . 15 
of the tooth in the < 

( pinion, a = 0-5 P(l — J?LY - 16 

A rack is to be considered as a wheel of 200 teeth. 



180 Gearing. 



Example witli Plate IV e 

Example. A wheel of D == 48 inches diameter is to gear a pinion about 8 
revolutions to 1. Required a complete construction of the gearing? 

Approximate pitch P = 0*028X48=1*34 in. - - - 7 
wheel, Jf- ?^i 8 = 112. . 8 



Number of teeth 
in the 



1 . . 112 

Opinion, m = —7T~ — 14 



Half the an- ( whee1 ' P " -^f^S^O-O^. 1 

gle between < 

tWO teeth in | . . yr ^® to t?^^ • noon/ n 

,i Opinion V= — jr— =12°51 . sm=0*2224. 2 

Diameter of pinion d = 48 n X '^ 8 = 6'043 in. - 4 
r 0*2224 

Draw the pitch circle for the wheel and pinion so that they tangent one an 
other at r on a straight line between the centres of the circles. 

Pitch in the gearing P = 48X0-028=1*344 in. - 5 

Take this chordial pitch in a pair of compasses, and set it off in the pitcii 
circles. 

( wheel a = 1*344 (o*42 + *i Y=0*592in. 15 
Thickness of 1 \ 700 / 

tooth J / 14 \ 

Opinion a - 0'5Xl«344rf 1-^1=0-645 in. 16 

Set off the thickness of tooth in the corresponding pitch circles. 
Bottom clearance b = 0*4X1-344 = 0*5376 in. - - 11 
Depth to pitch line c = 0*3 \ 1*344=0*4032 in. - - 12 

, 1*344(112+6) n .H Qrl • , Q 

Distances r o and f d " 2(112-11) " 13 

r o' in the wheel ( . „ . _ . . 

e = 0*llXl-344</112 = 0*7126 in. 14 

Set off these distances on the line o o' from r,—d beyond and e within the 
pitch circle for the wheel ; then o is the centre and o m radius for the flank m. 
o' the centre and o' n radius for the face n. Draw circles through o and o 7 con- 
centric with the pitch circle of the wheel. 



,„-i„th e pii,io« l <! , . llxl . 3U y n , . 36Sin . u 



Distances r o and f 2(14 — 11) 



Proceed with the pinion similar as the wheel 

On the plate is a scale of inches and decimals, which will be con- 
venient for the above measurements. 



(rearing. 



FlatelV. 




1=-- -q fif 



flat&V. 




Flanges & Stuffing Boxes. riateTi 




r 

Diameter 



J.W.Ny strom. 



Strength op Materials. 181 



Strength of Teeth. 

Letters denote, 

S — strain on the teeth at the pitch-line in pounds. 

a = thickness, (see figure),) ., 

h = breadth of the teeth, | mcnes - 

v = velocity of the teeth in feet per second. 

J3"= Horse power transmitted by the teeth. 

r a = Q'02tyS, r S = 1600 a*, 

Thickness < r-jr Strain 1 

la = 0-6634 /-, I S = 300 ft a. 



. p = 0-0544^, , H = 2-275 ^a r, 



ntch 1 /-g. Horses 



V? 






R=0'4 



Abrasion included in these formulas and the breadth h = 2*5p. 

"When great strain is required on the teeth, and the diameter or pitch of the 
■wheel is limited, it is necessary to increase the breadth h proportionally, which 
will be thus, 

_ S_ _ S Sm Hm H B 



' 653p 300a 943D 1*347 D v 0'429a v 0'934p v' 

Example. The pinion-wheel on a propeller shaft is to be D = 48 inches in 
diameter, and to have m = 36 teeth; it is driven by a pair of engines H = 450 
horses, and the propeller to make n = 50 revolutions per minute. Required 
the breadth of the teeth h = 1 The velocity at pitch circle will be, 

3'14X'4y50 
v = g^ = 10*5 feet per second, nearly, and 

h = i*s3r, = mffSxiw = 23 ' 88 ' or * inche *' neatlr - 

To Find the Diameter of Axles and Shaft. 

Letters denote, 
d = diameter in inches, in the bearing ; and the length of the bearing VSd. 
W= weight in pounds, acting in the bearing. 

d = 1— of cast iron. Common A = ~- of cast iron. 

Water- J lb Machinery) ** 

wheels. 1 -— in good 1 r^ 

^d = K _ of wrought iron. order, ^a *- f 

Example 1. A water wheel weighs 58,680 pounds, and supported in two bear- 
ings. Required the diameter of the wheel axles ? The weight acting in each 
bearing will be 58680 : 2 = 29340 pounds, and 

diameter d = — = 8*15 inches oi wrought iron. 

L 1 

Examptb 2. Fig. 226, page 185. Required the diameter of the axle in the 
wheel, when the weights P + Q = 4864 pounds? If the wheel is supported 
in two bearings TF= 4S64 : 2 = 2432 pounds. 

i 24-*-} 2 
diameter d = v — - = 1*76 inches of wrought iron. 

— 



182 



Strength of Materials. 



nrH%? mple ?* T ^ e P r . essure on the steam piston, in a walking beam engine is 
25000 pounds. Required the diameter of the beam journals ? 

diameter d = — 1|^_ = 5*64 inches the centre one. 

« ~~ ■ 2o ' = 4 mcnes a ^ the ends. 

In this example it is supposed that the beam is worked by a fork connecting 





D = 



\/FR 



=\7 



S 



D = inches wrought iron. 

R = radius of crank in feet. 

F = force from the steam piston, lbs. 



D:d = s/ R : &T~ 

H = horse-power transmitted. 

n = number revolutions per minute. 



"When an axle or shaft not only serves as a fulcrum, but effect is transmitted 
by the act of twisting it, the diameter is to be caluulated as follow. 

Example 4. The pressure on the piston in a steam engine is F=* 45,600 
pounds, applied direct on a crank of R = 3 feet radius. Required the diameter 
of the shaft and crank pin 1 



Diameter of the shaft D '■ 



Diameter of the crank pin d 



^45600 X3 =12 , 9inches< 
4 
_ \/45600 



28 



7*63 inches. 



Example 5. A steam engine of 3G8 horses is to make 32 revolutions per 
minute. Required the diameter of the main shaft? 

Diameter D = 5 \ / -J£- = ll i inches. 

Example 6. A cog wheel of E = 6*5 feet radius is to gear with a pinion of 
r = 1*25 feet radius, and to transmit an effect of 231 horses with 42 revolutions 
per minute. Required the diameter of the wheel and pinion shafts 1 The force 
Fis acting uniformly at the periphery. 

8 / 231 

Diameter of wheel shaft D = 4*35 \ / - An . = 7'66 inches 

V 42 

D:d= V~Tf 
Diameter of pinion shaft d = 7*66 



V r 
3 



V 6-5 



4-41 inches. 



ALLOYS. 

Letters denote. 

A = Antimony, B = Bismuth, (7 = Copper. G = 
2V= Nickel, S= Silver, T= Tin, and Z = Zinc. 
Brass, yellow, .... - 

" rolled, ------- 

Brass-casting, common, - - - - 

" hard, ------ 

Brass-Propellers, (large), - - - - - 

Gun-metal, -------- 

Copper-flanges, ,/br pipes, - - - - - 

Brass that bears soldering well, - 

Muntz's metal can be rolled and worked at red heat, 6(7, 4Z. 



183 



Gold, I = Iron, L =s Lead, 

20, \Z. 

32 C, 10Z, 1-5 T. 

20 C, 1-25Z, 2-521 

25(7, 2Z, 4-52: 

8(7, 0-5Z, IT. 

8(7, IT. 

9(7, \Z, 0-26T. 

20, 0-15Z. 



91-4(7, 5-53Z, 1-7 7, 1-37X. 

20 O, 15-8iV. 12-7Z, 1-3J. 

53-39(7, 17-42V. 13Z. 

100 O, SZ. 

5(7, 1Z, 

65-2(7, 19-5Z, 13iV, 2-5S, 12 
cobalt of I. 

Britannia metal, \Z, \A \ . ~ „ . * D 

When fused add, lA,lBf 1Z > ZA > 1B ' 

Babbitt's anti- Attrition metal, - 25T,2A0'5O. 

m The Tin of the best quality of Banca, is to be added gradually to the melted compo- 



Statuary, 
German Silver, - 
Frick's Imitative Silver, 
Medals, - 
Pinchbeck, - 
Chinese Silver, - 



Bell-metal, large, 

" small, 

Gold Metal. 

x = 21(7 13 T ) 

y = 620 9Z j To ^ e melte< * separately. 

Gold = 71y+9x, this makes a brilliant composition. 



3(7, IT. 
4(7, IT. 



Newton's fusible alloys, 
Rose's " " 

A more fusible composition 
Tin solder, coarse, 

ordinary, 
Soft Spelter-solder, for common 
Hard " for iron, 

Solder for Steel, 
Solder for fine brass works, 
Pewterer's soft solder, - 
it a 

Gold Solder, 
Silver solder, hard, 

toft, - 



Solders. 



SB, 5Z, ZT, melts at 212°. 
2B, 1L, IT, « 201o. 

bB; ZL, 2T, « J 99°. 

IT, ZL, «• 500°. 

2T, IX, " 360°- 

brass works, 1(7, 1Z. 
2(7, 1Z 
19S, 3(7, 1Z. 
IS, 8(7, 8Z. 
2JS, 4£, ZT. 
IB, 1L, 2T 
24(7, 2& 1(7 
4& 1(7. 
2£, 1 brass wire. 



Tempering Steel* 

Tern. Fah. 

Yellow, very faint, for lancets - 4300 

„ pale straw, for razors scalpels - 4500 

„ full, for penknives and chisels for hard cast iron - 470° 

Brown, for scissors and chissels for wrought iron - - 490° 

Red, for carpenter tools in general - 510° 

Purple, for fine watch springs and table knives - - 530° 

Blue, bright, for swords, lock springs - 5500 

„ full, for daggers, fine saws, needles - - 560° 

„ dark, for common saws - 600° 



184 



Weight op Rolled Iron, per Foot. 




ft 



i 

H 
li 
if 

14 
if 
if 
« 

2 

24 

2i 

2f 

24 

2£ 

21 

2* 

3 

34 

3i 

31 

3* 



Weight in 


Side in 


pounds 


inches. 


0-013 


3t 


0-53 


31 


0-118 


3£ 


0-211 


4 


0-475 


4* 


0-845 


4£ 


1-320 


« 


1-901 


44 


2-588 


4* 


3-380 


4| 


4-278 


4£ 


5-280 


5 


6-390 


5* 


7-604 


5£ 


8-926 


5ft 


10-325 


54 


11-883 


5* 


13-520 


51 


15-263 


53 


17-112 


6 


19-066 


« 


21-120 


6* 


23-292 


61 


25-56 


7 


27-939 


n 


30-416 


8 


33-010 


84 


35-704 


9 


38-503 


10 


41-408 


12 



Weight in 
pounds 



44-418 
47-534 
50-756 
54-084 
57-517 
61-055 
64-700 
68-448 
72-305 
76-264 
80-333 
84-480 
88-784 
93-168 
97-657 
102-24 
106-95 
111-75 
116-67 
121-66 
132-04 
142-82 
154-01 
165-63 
190-14 
216-34 
244-22 
273-79 
337-92 







1 

H 
li 

it 
14 

il- 
ia 
i* 

2 

24 

2i 

21 

24 

2ft 

21 

23 

3 

34 

3* 

3ft 

34 



Weight in 


Diameter 


pounds. 


in inches. 


o-oio 


3ft 


0-041 


31 


0-119 


31 


0-165 


4 


0-373 


44 


0-663 


4i 


1-043 


4f 


1-493 


44 


2-032 


4ft 


2-654 


4f 


3-360 


4£ 


4-172 


5 


5-019 


5| 


5-972 


5i 


7-010 


51 


8-128 


54 


9-333 


5ft 


10-616 


55 


11-9^8 


5$ 


13-440 


6 


14-975 


61 


16-688 


64 


18-293 


61 


20-076 


7 


21-944 


74 


23-888 


8 


25-926 


84 


28-040 


9 


30-240 


10 


32-512 


12 



Weight in 
pounds. 

34-886 
37-332 
39-864 
42-464 
45-174 
47-952 
50-815 
53-760 
56-788 
59-900 
63-094 
66-752 
69-731 
73-172 
76-700 
80-304 
84-001 
87-776 
91-634 
95-552 
103-70 
112-16 
120-96 
130-05 
149-33 
169-85 
191-81 
215-04 
266-29 
382-21 



Cements for Cast Iron* 

Two ounces Sal-ammoniac, one ounce Sulphur and sixteen ounces of 
borings or filings of cast Iron, to be mixed well in a mortar, and kept dry. 
When required for use take one part of this powder to twenty parts of clear 
iron borings or filings, mixed throughly in a mortar, make the mixture into a 
stiff paste with a little water, and then it is ready for use. A little fine grindstone 
sand improves the cement. 

Or one ounce of Sal-ammoniac to one hundred weight of Iron borings. No 
heat allowed to it. 

The Cubic contents of the joint in inches, divided by five, is the weight of dry 
borings in pounds Avoir, required to make cement to fill the joint nearly. 
Cement for Stone and Brick work* 

Two parts Ashes, three of Clay, and one of Sand, mixed with oil, will resist 
weather equal to marble. 

Brown Mortar. 

One part Thomaston lime, two of Sand, and a small quantity of Hair. 
Hydraulic Mortar. 

Three parts of Lime, four Puzzolana, one Smithy Ashes, two of Sand, and four 
parts of rolled stone or shingles. 





Weight 


of Riveted Pipe3, per Foot 


tiie Laps 


INCLUDED. 




185 


Diameter 


Thickness 


Copper pipe 


Iron pipe 


Diameter Thickness 


Copper pipe 


Iron pipe 


inches. 


16th in. 


pounds. 


pounds. 


iuches. 


16th in. 


pounds. 


pounds. 


5 


3 


12-50 


10-96 


n 


4 


30-59 


26-81 


5 


4 


16-88 


14-78 


10 


4 


32-21 


28-21 


51 


3 


13-15 


11-52 


10^ 


4 


33-94 


29-70 


51 


4 


17-75 


15'55 


11 


4 


35-20 


30 84 


54 


3 


13-63 


11-94 


114 


4 


36-94 


32-35 


54 


4 


18-39 


16*07 


12 


4 


38-45 


33-67 


55 


3 


14-25 


12-48 


13 


4 


41-45 


36 30 


5| 


4 


19-25 


16-86 


14 


4 


44-64 


39-11 


6 


3 


14-76 


12-94 


14 


5 


55-88 


48-97 


6 


4 


19-91 


17*43 


15 


4 


47-64 


41-74 


61 


3 


15-36 


13-46 


15 


5 


59-59 


52-20 


61 


4 


20-75 


18-16 


16 


4 


50-75 


44-45 


6^ 


3 


15-90 


13-93 


16 


5 


63-47 


55-60 


64 


4 


21-41 


18-75 


17 


4 


53-86 


47-15 


61 


3 


16-50 


14-45 


17 


5 


67-34 


59-00 


61 


4 


22-25 


19-70 


18 


4 


57-04 


50-00 


7 


3 


17-03 


14-93 


18 


5 


71-26 


62-41 


7 


4 


22-93 


20-07 


19 


4 


60-14 


52-65 


71 


3 


17-65 


15-45 


19 


5 


75-23 


65-90 


71 


4 


23-74 


20-79 


20 


4 


62-51 


54-74 


n 


3 


18-32 


16-05 


20 


5 


78-21 


68-5 


n 


4 


24-45 


21-40 


21 


5 


82-98 


72-64 


n 


3 


18-95 


16-60 


22 


5 


86-77 


76-00 


71 


4 


25-28 


22-13 


23 


5 


90-57 


79-34 


8 


3 


29-42 


17-03 


24 


5 


94-31 


82-60 


8 


4 


25-96 


22-72 


26 


5 


101-9 


89-32 


Sh 


3 


20-58 


18-03 


28 


5 


109-4 


95-68 


84 


4 


27-47 


24-07 


30 


r* 


117-0 


102-4 


9 


4 


28-98 


25-38 


36 


5 


140-0 


122-5 




Weight of Cast Iron 


Cylinders per 


Foot. 


Diameiei 


Weight 


hi'imtt-ir 


Weight 


Diameter. Weight 


Diameter 


Weight 


in 


in 


in 


in 


in 


in 


in 


in 


inches. 


pounds, 


inches. 


pounds. 


inches. 


pounds. 


inches 


pounds. 


ii 

4 


1-39 


n 


18-74 


4| 


55-92 


74 


139-4 


I 


1-88 


n 


20-48 


4£ 


58-72 


71 


148-87 


lin. 


2-47 


3in. 


22-35 


5in. 


61-96 


8in. 


158-63 


H 


3-13 


n 


24-20 


8i 


64-66 


81 


168-15 


H 


3-87 


31 


26-18 


51 


68-31 


84 


179-1 


li 


4-68 


3§ 


28-23 


5| 


71-00 


81 


189-0 


14 


5-57 


34 


30-36 


54 


74-98 


9 in. 


200-8 


11 


6-54 


31 


32-57 


5| 


78-65 


91 


211-12 


If 


7*59 


3| 


34-85 


5S 


81-95 


94 


223-7 


11 


8-7-1 


35 


37-21 


H 


85-81 


9| 


235-3 


2in. 


9-91 


4in. 


39-66 


6in. 


89-23 


10in. 


247-9 


14 


11-19 


4* 


41-80 


61 


96-82 


101 


273-27 


21 


12-54 


41 


44-77 


64 


104-7 


11 in. 


299-9 


23 


13-98 


4| 


47-00 


6| 


112-9 


114 


327-8 


24 


15-49 


4^ 


50-19 


7in. 


112-4 


12in. 


356-9 


U 


17-08 


4f 


52-71 


71 


130-28 


13in. 


418-9 



16* 



ISO 




Weight of Cast Iron Pipes per Foot. 






ThicJcness of Metal. 






Itora. 


i 1 


* 


1 


I 2 


! I 


! 1 


1 


3.06 


5'05 








i 




1* 


3'67 


6-00 












1* 


6*89 


6'S9 


9-81 










11 




7-80 


11-04 










2 




8*74 


12-23 


16-0 








2i 




9*65 


13-48 


17-52 








2* 




10*57 


14-66 


19-05 


23-8 






2| 




11*54 


15-91 


20-59 


2568 






3 




12-28 


17-15 


22-15 


,27*56 


S3-30 


39-31 


3| 




13-24 


18-40 


23-72 


29-64 


35-46 


41-77 


3* 




14-20 


19-66 


25-27 


31-20 


37-63 


44-23 


3| 




15*50 


20-90 


26-83 


33-07 


39-77 


46-68 


4 




16*80 


22-05 


28-28 


34-94 


41-92 


49.14 


4* 




17-41 


23-35 


29-85 


36-73 


44-05 


51-57 


4* 




18-00 


24-49 


31-40 


38-58 


46-19 


54-00 


4| 




18'89 


25-70 


32-91 


40-43 


48-34 


56-45 


5 




19-79 


26-94 


34-34 


42-28 


50-50 


58-90 


5* 




21-54 


29-40 


37-44 


45*94 


54-81 


63-82 


6 




23-42 


31-82 


40-56 


4960 


58-96 


68-70 


6| 




25*26 


34-32 


43-68 


53-30 


63-18 


73-40 


7 




27*15 


36-66 


46-80 


56-96 


67-60 


78-39 


7* 




28*92 


39-22 


49-92 


60-48 


71-76 


83-28 


8 




30-76 


41-64 


52-68 


64-57 


76-12 


88-20 


8* 




32-82 


44-11 


56-16 


68-00 


80-50 


73-28 


9 




34-45 


46-50 


58-92 


71'7C 


84-70 


97-98 


n 




36-26 


48-98 


62-02 


75-32 


88-98 


102-9 


10 




38-15 


54-46 


65-08 


78-99 


93-24 


108-8 


io| 






53-88 


68-14 


82-63 


97-44 


112-7 


n 






56-34 


71-19 


86-40 


101-83 


117-6 


Hi 






58-82 


74-28 


90-06 


106-1 


122-6 


12 






61-26 


77-36 


93-70 


110-5 


127-4 


12* 






63-70 


80-40 


97-40 


114-7 


132-5 


13 






66-14 


83-46 


101-1 


11S-9 


137-3 


13* 






68-64 


86-55 


104-8 


123-3 


137-28 


14 






71-07 


89-61 


108-46 


127-6 


147-0 


14* 






73-72 


92-66 


112-1 


131-9 


151-9 


15 






75-96 


95-72 


115-8 


136-2 


156-8 


15* 






78-40 


98-78 


119-5 


140-4 


161-8 


16 






80-87 


101-8 


123-1 


144-8 


166-6 


16* 






83-30 


104-8 


126-8 


149-0 


171*6 


17 






85-73 


107-9 


130-5 


135-3 


176-6 


17* 






88-23 


111-1 


134-2 


157-6 


181-3 


18 








114-1 


137-8 


169-9 


186-2 


19 








120-2 


145-2 


170-5 


195-9 


20 








126-3 


152-5 


179-0 


205-8 


21 








132-5 


159-8 


187-6 


215-5 


22 








138-6 


167-2 


196-5 


225*4 


23 








144-8 


147-6 


204-8 


235-3 


24 








150-8 


181-9 


213-28 


245-1 


25 










196-6 


230-6 


264-7 


28 










211-3 


247-6 


284-3 


30 










226-2 


264-8 


303-9 



Weight of Flat Rolled Iron per Foot. 187 



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188 




Weight of Flat Rolled 


Iron per 


Foot. 


















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W 







"Weight 


of Flat Roli 


KP 


Iron per 


FOOT. 












189 






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rU 


o 


cb 


o 


ib 


CO 


rH 


O 


rH 


J>= 


■CO' 


op 


iO 


CO 




u 

H 


rd 


CO CO CM CM 


CM CM 


cq 


CM 


rH 


r-' 


rH 


rH 


rH 


r-i 


cb 


cb 


o 


co 


eq 


rH 


rH © b- OS 


iO CM 


:•: 


-H 


O 


x^ 


rH 


OS 


CO 


CM 


CO 


b- 





rH 


10 


b- 


^3 

d 

c3 


*fao o° ^ *? T 1 

CO CO OS b- CO 


ip OS 


CM 


SC 


CO 


op 


.t- 


o 


■rH 


O^ 


CO 


r J 




b- 


10 


CO 


rH cm 




OS 


00 


cb 


A 


CO 


H 


oo 


7^ 


iQ 


CO 


CM 


rH 


to 


»e 


CO CM CM CM 


CM CM 


CM 


T— J 


— 1 


rH 


rH 


rH 


—i 


OS 


CO 


cb 


rH 


cb 


CM 


rH 


OS rH CO iO 


CO CO 


OS 


^— 1 


C>5 


~ 


-b- 


b- 


oa 


>ra 


rH 


CO 


o^ 


GO 


O 


rH ' 


9 


a 

^d 


^ f5 f5 ^ M 
CO OS CO CO o 


.b- rH 


o 


CO 


rh 


bo 


CM 


o 


o 


CO 


CM 


CO 


iO 


CO 


b- 


CO 


1-4 


CO CM 


o 


Cb; 


b- 


b> 


rH 


CC3 


J_i 


ip 


oa 


co 


b- 


7H 


eo 


ip 





N M N N 


CM CM 


cq 














cb 


b- 


cb 


■H 


CO 


C>5 


rH 


rH © OO © 


b- H< 


o 


b- 


^ 


rH 


OS 


CO 


CM 


oo 


b- 


-0 


rH 


CM 


b- 


rH 




^ CO o b- »p 

CO CO N iO T^ 


OS rH 


oa 


op 


CO 


op 


Jb- 


irq 


b- 


oo 


iO 


0; 


CO 


CO 


co 


CO 


•M 


CM rH 


OS 


CO 


cb' 


ib 


cb 


05 


O 


7H 


CO 


7H 


iQ 


CO 


CM 


ip 


S3 




N N (N IN 


CM CM 



















b- 


CO 


rH 


CO 


CM 


rH 


OS T— 1 CO UT5 


CO o 


oq 


rH 


SO 


oo 


— I 


CO 


o 


rH 


co 


rH 


CO 


b- 


00 


OS 


«H 


^v- © CO tH CO 

CO CO CO iO CO 


T 1 ^" 


CM 


fc- 


CM 


b- 


CO 


co 


CO 


J>- 


CS 




CO 


10 




i^ 





cd 


CM © 


OS 


b- 


cb 


rH 


■cb 


rH 


o 


O0 


co 


co 


H^ 


CO 


« 


r}< 


CD 

,d 


CM CM CM CM 


CM CM 
















00 


b- 


ib 


rH 


0^ 


b; ; 


rH 


CO CO CO i—l 


CO CO 


CO 


o 


CO 


iO 


CO 


CO- 


o 


rH 


05 


CO 


b- 


rH 


00 


CO 


•w 


+3 


^ tH CO CM CO 
CO b- m ■*& CM 


CO OS 


o 




CO 


CM 


oo 


^ 


CO 


i« 


CM 





b- 


O 


co 


CM 


£ 


.9 


rH OS 


oo 


-b- 


ib 


A 


bi 


rH 


cs 


>p 


7H 


b- 


CM 


00 


rH 


rH 


if 


CM CM CM CM 


CM rH 














cb 


00 


b- 


uo 


rH 


cfl 


b^ 


rH 


OO rH rH b- 


OS CM 


lO 


fr- 


o 


co 


CO 


oo 


o 


b- 


uO 


oq 


00 


O 


00 


CO 




^ O N CO OS 
CO © rH CO rH 


ip CM 


00 


T* 1 




b- 


op 


cs 


rH 


co 


CO 


OS 


pH 


rH 





b- 


xn 


© 6s 


b- 


o 


»b 


cb 


oq 


o 


cp 


CM 


00 


rH 


r~ 1 


b- 





CO 


w 


CM CM CM CM 


CM rH 


--' 


i — i 


T— 1 


r— 


rH 


r- 1 


cb 


00 


<b 


ib 


rH 


<fq 


CM 


rH 


b- CO rH CM 


© CO 


eo 


-H 


CM 


CO 


:' 


O 


o 





r-i 








CO 


O 


CO 




^ C N "* H 
CO iO CO CM rH 


CO rH 


rH 


co 


O 


CM 


co 


ip 


rH 


CM 


CO 


00 


CO 


rH 


CO 


CM 




OS CO 


t-m 


>b 


A 


M 


^1 


o 


b- 


co 


CO 


CM 


CO 


CO 


CO 


CO 




CM CM CM CM 
















O 


b- 


cb 


ib 


cb 


be; 


rH 


rH 


b- r- 1 Hi CO 


rH rH 


b« 


rH 


rH 


b- 


rH 


rH 


-H 


rH 


j>= 


CO 


CM 


iQ 


iH 


b- 




CO CO iO CM 


O b~ 


rH 


tq 


Oa 


CO 


r^ 


rH 


b- 


O 


CO 


cO 


O 


CO 


O 


CO 




*d 
H 


M ^ N H O 


6s b- 


cb 


ib 


CO! 


CM 


rH 


O 


op 


ep 


CO 


O 


00' 


ip 


CO 


CM 




CM CM CM CM 


r-i r-i 


Hi 


r— l 


rH 


iH 


rH 


rH 


CO 


b. 


cb 


-ib 


CO 


CM 


^ 


rH 


t- © O CO 


CM O 


oa 


J^ 


O 


rH 


CO- 


CO 


CM 


b- 


cq 


oo 


— 


CO 


CM 


iO 






H» CO CO CO rH 


CM CO 


b- 


O 


op 


rH 


OS 




© 


00 


b- 


10 


r* 


CM 


07 








N CO rH © OS 


CO J>- 


o 


rf) 


cb 


i?q 


o 


b- 


O 


ccq 


co 


00 


CO 


rH 


OO 


CM 






CM CM CM tH 














cb 


00 


b- 


cb 


rH 


cb 


CM 


iH 


rH 


b~ rH iO CO 


c<« o 


o 


"* 


00 


CM 


o 


rH 


CM 





00 


b- 


iQ 


CO 


— 


CM 






KK O © N >0 


"? ^ 




oa 


b- 


O 


rH 


o> 


CO 


b- 


O 


rH 


OO 


05 


rH 


CO 






^ CM O OS CO 


j>- cb 


^b 


CO 


c>q 


rH 


cb 


CM 


rH 


oa 


00 


CO' 


rH 


op 


b- 


7H 






CM CM tH tH 


rH r-t 


rH 


rH 


rH 


rH 


rH 


■cb 


cb 


cb 


ib 


rH 


cb 


<cb 


rH 


rH 


i>. CO iO rH 


CO CM 


CM 


_i 


O 


crs 


1 — | 


CM 


c* 


-H 


iO 


CO 


CM 


oc 


CO 


OS 






^ © OS CO b- 
CM H OS CO b- 


CO iO 


H-, 


00 


CM 


o> 


CO 


b- 


CO 


[O 


rH 


CO 


b- 







O 






co ib 


■^ 


cb 


CM 


rH 


oa 


CO 


^" 


CO 


ifj 


rH 


eo 


CM 


CO 


rH 






CM rH rH tH 












cb 


cb 


b* 


cb 


uo 


rH 


CO 


bi 


rH 


rH 


CO rH iO OS 


~* CO 


CO 


X— 


CM 


CO 


-H 


00 


CM 





CO 


rH 


CO 


CC-5 


rH 


CO 






^ © © os cp 

CM © OS b- CO 


CO b- 


J^ 


CO 


O 


*? 


o 


•H 


•OS 


co 


CO 


CM 


CO' 




CO 


iO 






ib 4n 


cb 


CM 


rH 


o 


iO 


rH 


CO 


CO 


CM 


CM 


rH 


7H 


Ip 









CM rH rH rH 












cb 


00 


b- 





ib 


rH 


cb 


CM 


rH 


rH 


e*D H-$ HpO 


H=o w 


■efeo 


-■^ 


^?X 


i-W 


r*B 




H« 


«^ 


■4a 


r-tn 


•;-,^ 


rH 








<N CN C* (N 





























190 



V'ETGTIT r,F MATERIALS. 



Weight Per Square Foot in Pounds. 



Tfaickoaa 












in iochti. 


Cast Iron. 


Sheet Iron. 


Sheet Copper. 


6heet Lead. 


fhoet Zir.c 


1 
16 


2-346 


2-517 


2-890 


3-694 


2-320 


k 


4-693 


5-035 


5-781 


7-382 


4-642 




7-039 


7*552 


8-672 


11-074 


6-961 


1 


9-3S6 


10-070 


11-562 


14-765 


9-275 


7T 
T6 


11-733 


12-588 


14-453 


18-456 


11.61 


§ 


14-079 


15-106 


17-344 


22-148 


13-93 


7 


16-426 


17-623 


20-234 


25-839 


16-23 


* 


1S-773 


20 141 


23-125 


29-530 


18-55 


9 

TB~ 


21-119 


22-659 


26-016 


33-222 


20-87 


1 


23-466 


25-176 


28-906 


36-913 


23-19 


is 1 


25-812 


27-094 


31-797 


40-604 


25*53 


i 


28-159 


30-211 


34-6SS 


44-296 


27-85 


fl 


30-505 


32-729 


37-578 


47-987 


30-17 


i 


32-852 


35-247 


40-469 


51-678 


32-47 


» 


35-199 


37-764 


43-359 


55*370 


34-81 


i 


37-545 


40-2S2 


46-250 


59-061 


37-13 


14 


42-238 


45-317 


52-031 


66-444 


41-78 


11 


46-931 


50-352 


57-813 


73-S26 


46-42 


if 


51-625 


55-387 


63-594 


63-594 


51-04 


l* 


56-317 


60-422 


69-375 


88-592 


55-48 


if 


61-011 


65-453 


75-156 


95-975 


60-35 


if 


65-704 


70-493 


80-938 


103-358 


65.00 


if 


70-397 


75-528 


86-719 


110-740 


69-61 


2 ! 


75-090 


SO-563 


92:500 


118-128 


74-25 



Weignt of Copper Rods or Bolts per Foot, 



Diameter. 


Weight. 


Diameter. 


Weight. 


Diameter. 


Weight. 


Diameter 


Wei*M. 


Inches. 


Pounds. 


Inches 


Pounds. 


Inches. 


Pounds. 


Inches. 


Pounds. 


JL 

4 


0-1892 


l 


3-0270 


lj 


10-642 


3f 


34-487 


"5" 
TB~ 


0-2956 


V. 


3-4170 


2 


12-108 


31 


37-081 


t 


0-4256 


1* 


3-8912 


2£ 


13-668 


3| 


39-737 


"7" 


0-5794 


r ?« 


4-26S8 


n 


15-325 


31 


42.568 


* 


0-7567 


n 


4-7298 


21 


17-075 


3£ 


45-455 


9 

TB" 


0-9578 


W» 


5-2140 


21 


18-916 


4 


48-433 


i 


1-1824 


j» 


5-7228 


21 


20-856 


41 


53-550 


tt 


1-4307 


1 T6 


6-2547 


2| 


22-891 


41 


61-321 


f 


1-7027 


li 


6-8109 


n 


25-019 


41 


68-312 


\l 


1-9982 


It 9 ,, 


7*3898 


3 


27-243 


5 


76-130 


3 


2-3176 


IS 


7-9931 


n 


29-559 


51 


91-550 


l 5 

16 


2-6605 


n 


9-2702 


H 


31972 


6 


109- 



lUrmiiigham Gauge for Wlrc» Sheet Iron and Steel* 101 





Weight per Square 


Foot in Pounds. 




Thickness by 


Thickess in 


Sl'cet and 


Sheet Cast 


Sheet 


Thickness 


iht: Gau^e. 


Inches. 


Boiler Iron. 


. Steel. 


Copper. 


in Inches. 


INo. 


0-340 


13-7 


14-0 


15-6 


H 


" 1 


0-300 


12-1 


12-4 


13-8 


1 6 


" 2 


0.284 


11-4 


11-7 


13-0 


9 
32" 


'• 3 


0-259 


10.4 


10-6 


11-9 


i 


« 4 


0-238 


9-60 


9 80 


110 


A 


14 5 


0-220 


8-85 


■ 9-02 


10-1 


a 


" 6 


0-203 


8-17 


8-33 


9-32 


u 


" 7 


0-180 


7*24 


7*38 


8-25 


/. 


" 8 


0-165 


6-65 


6-78 


7-59 


•< 


" 9 


0*148 


5-96 


6-08 


6-80 


5 

5 2" 


"10 


0-134 


5-40 


5-51 


6-16 


u 


"11 


0-120 


4-83 


4-93 


5-51 


i 


"12 


0-109 


4-40 


4-50 


5-02 


cc 


"13 


0-095 


3-83 


3-91 


4-37 


A 


"14 


0-083 


3-34 


3-41 


3-81 


It 


"15 


0-072 


2-90 


2-96 


3-31 


1 

16 


"16 


0-065 


2-62 


2-67 


3-00 


tt 


"17 


0-058 


2-34 


2-39 


2-67 


u 


"13 


0-049 


1-97 


2-01 


2-25 


u 


'19 


0-042 


1-69 


1-72 


1:93 


2 
65 


"20 


0-035 


1-41 


1-42 


1-61 


a 


"21 


0-032 


1-29 


1-31 


1-47 


a 


"22 


0-028 


1-13 


1-15 


1-29 


sV 


"23 


0-025 


1-00 


1-02 


1-14 




" 24 


0-022 


0-885 


0-903 


1-01 


(t 


"25 


0-020 


0-805 


0-820 


0-918 


« 


"26 


0-018 


0-724 


0-738 


0-826 


eh 


"27 


0-016 


0-644 


0-657 


0-735 


a 


"28 


0-014 


0-563 


0-574 


0-642 




" 2 9 


0-013 


0-523 


0-533 


0-597 




"30 


0-012 


0-483 


0-493 


0-551 




31 


o-oio 


0-402 


0-410 


0-480 




3 2 


0-009 


0-362 


0-370 


0-420 




"33 


0-008 


0-322 


0-328 


9-370 




" 34 


0-007 


0-282 


0-288 


0-323 




"35 


0-005 


0-230 


0-235 


0-262 




"36 


0-004 


0-170 


0-173 


0-194 






A new wire gauge has been started by Brown & Sharpe, of Providence, 
R. L, and called the " American Standard wire gauge." It is an im- 
provement on the Birmingham gauge, but still it must be accompanied 
by an interpreter to explain what it is. Not one in a hundred or one in 
a thousand of those who have to deal with measures, would understand 
the thickness from the number of the wire gauge. Whenever it is 
written or spoken of, it must be translated into inches in order to make 
it clear how much it is ; why not then have the numbers of the wire 
gauge expressed direct into inches as proposed by M. Whitworth a few 
years ago, published I believe in the Artizan for 1857. If the American 
manufacturers would take up the proposition of M. Whitworth it would 
be one step ahead. It is very clear that the wire gauge is patched up 
from different sources and ages to its present state, which is not worth 
to imitate in a new guage. 



Proportion of Bolts and Nuts* Number of threads per In* 



Diameter of Bolt. 



a l k- 






m 




n*^<o h. *. oh m h m £.£ £ £ *>£ ■££& W »S2 & fc. ^ 



o 



MMWWWWMM^^OiCnOS ~J 




►-« H-» i_i >-» »-J MHWM ^ Air 

*^ H,H, M ^H ^ ^ ^ <tf* ^ o^ o** M 1 -^ of- *+- °*-> »*- a*« **» **-»*- 




* til 









Table 


for Falling Bodies 






193 


Velo- 
city 

ntthe 


1 

! Space fall- 


1 

Time in 


Velocity 


1 i 

Space fall- ! Time in 


Vcjoci ry 


1 

Space fall. 


Time in 


en through 


seconds. 


at i he 
eud. 


en through 


seccude. 


at the 

eud. 


eu through 


secouds. 


V 


s 


T 


V 


s 


T 


V 


s 


T 


0-1 


•00015 


0-0031 


5-1 


•40388 


0-358 


11 


1-8789 


Q-342 


0.2 


•00062 


0*0062 


5-2 


•41987 


0-162 


12 


2-0652 


0-373 


0-3 


•00139 


0-0093 


5-3 


•43618 


0-165 


13 


2-6242 


0-405 


0-4 


•00248 


0-0124 


5-4 


•45279 


0-168 


14 


3*0435 


0-436 


0-5 


•00383 


0-0155 


5-5 


•46972 


0-171 


15 


3-4938 


0-467 


0-6 


•00559 


0-0187 


5-6 


•48695 


0-174 


16 


3-9751 


0-498 


0-7 


•00761 


0-0218 


5-7 


•50450 


0-177 


17 


4-4876 


0-530 


0-8 


•00994 


0-0230 


5-8 


•52236 


0181 


18 


5-0310 


0-560 


0-9 


•01257 


0-0280 


5-9 


•55057 


0-184 


19 


5-6056 


0-591 


1- 


•01552 


0-0311 


6- 


•55900 


0-187 


20 


6-2112 


0-622 


11 


•01879 


0-0342 


6-1 


•57779 


0-190 


21 


6-8478 


0-654 


1-2 


•02065 


0-0373 


6-2 


•59689 


0-193 


22 


7-5155 


0-685 


1-3 


•02624 


0-0404 


6-3 


•61630 


0-196 


23 


8-2143 


0-716 


1-4 


•03043 


0-0436 


6.4 


•63602 


0-199 


24 


8-9441 


0-747 


1-5 


•03493 


0-0467 


6-5 


•65606 


0-202 


25 


9-7049 


0-778 


1-6 


'03975 


0-05 


6-6 


•67639 


0-205 


26 


10-497 


0-810 


1-7 


•04487 


0-052 


6:7 


•69705 


0-209 


27 


11-320 


0-840 


1-8 


•05031 


0-556 


6.8 


•71801 


0-212 


28 


12.174 


0-872 


1-9 


•05605 


0-0591 


6-9 


•73928 


0-215 


29 


13-059 


0-903 


2- 


•06211 


0-0623 


7- 


•76087 


0-213 


30 


13-975 


0-933 


2-1 


•06847 


0-0654 


7.1 


•78276 


0-221 


31 


14-922 


0-965 


2-2 


•07515 


0-0685 


7-2 


•80497 


0-224 


32 


15-900 


0-996 


2-3 


•0S214 


0-0717 


7-3 


•82748 


0-227 


33 


16-910 


1-025 


2-4 


•08944 


0-0747 


7-4 


•85031 


0-231 


34 


18-789 


1-058 


2-5 


•09705 


0-0780 


7-5 


•87344 


0-234 


35 


19-022 


1-091 


2-6 


•10497 


0-0810 


7-6 


•89689 


0-237 


36 


20-124 


1-120 


2-7 


•11320 


0-0841 


7-7 


•92065 


0-240 


37 


21-258 


1-151 


2-8 


•12174 


0-0872 


7-8 


•94472 


0-243 


38 


22-422 


1-184 


2-9 


•13059 


0-0903 


7-9 


•96910 


0-246 


39 


23-618 


1-213 


3- 


•13975 


0-0934 


8- 


•99379 


0-250 


40 


24-844 


1-243 


3-1 


•14922 


0-0966 


8-1 


1-0187 


0-253 


41 


26-102 


1-276 


3-2 


•15900 


0-0997 


8-2 


1-0441 


0-256 


42 


27-391 


1-308 


3-3 


•16910 


0-1025 


8-3 


1-0697 


0-259 


43 


28-57 


1-338 


3-4 


•18788 


0-1059 


8.4 


1-0956 


0-262 


44 


30-062 


1-370 


3-5 


•19022 


0-1092 


85 


1-1218 


0-265 


45 


31-444 


1-400 


3-6 


•20124 


0-1121 


8.6 


1-1484 


0-268 


46 


32-857 


1-431 


3-7 


•21257 


0-1152 


,8-7 


1-1753 


0-271 


47 


34-301 


1-463 


3-8 


•22422 


0-11S5 


8-8 


1-2015 


0-274 


48 


35-776 


1-495 


3-9 


•23018 


0-1214 


8-9 


1-2299 


0-278 


49 


37-282 


1-525 


4- 


•24844 


0-1246 


9- 


1-2577 


0-281 


50 


38-820 


1-555 


4*1 


•26102 


0-1278 


9-1 


1-2858 


0-283 


51 


40-388 


1-588 


4-3 


•27391 


0-1309 


9-2 


1-3143 


0-287 


52 


41-987 


1-619 


4-3 


•28571 


0-1339 


9-3 


1-3430 


0-290 


53 


43-618 


1-650 


4-4 


•30062 


0-1371 


9-4 


1-3720 


0-293 


54 


45-279 


1-680 


4-5 


•31444 


0-1403 


9-5 


1-4041 


0-296 


55 


46-972 


1-711 


4-6 


•32357 


0-1433 


9-6 


1-4310 


0-300 


56 


48-695 


1-742 


4-7 


•34301 


0-1465 


9-7 


1-4610 


0-302 


57 


50-450 


1-774 >. 


4.8 


•35776 


0-1496 


9-8 


1-4913 


0-306 


53 


52-236 


1-805 


4-9 


•37282 


0-1526 


9-9 


1-5219 


0-309 


59 


55-058 


1-835 


5- 


•38820 


0-1559 


10 


1-5528 


0-312 


60 ! 


55-900 


1-868 



194 



Gravitation. 



GRAVITATION. 



Gravity or Gravitation is a mutual faculty which all bodies in nature 
possess, to attract one another ; or Gravity is the force by which all bodies 
tend to approach each other. A large body attracting a comparatively very 
small one, and their distance apart being inconsiderable, the force of gravity in 
the small body will be very sensible compared with that in the large one ; such 
is the case with the body, our earth, attracting small bodies on or near her sur- 
face. 

Gravitation is not periodical, it acts continually ever and ever. A body placed 
unsupported at a distance from the earth, the force of gravity is instantly oper- 
ating to draw it down, and then we say, " the body fell down " If it were possi- 
ble to vWithdraw the attraction between the body and the earth, it would not 
fall down, but remain unsupported in the space where it was placed ; — giving 
the body a motion upwards it would continue that, and never come back to the 
earth again. 

Ijaw of Gravity* 

The force of Gravity is proportional to the mass of the attracting bodies, and in- 
verse as the square of their distance apart. 

This law was discovered by Sir Isaac Newton. It is this law that supports the 
condition of the whole universe, and enables us to calculate the distances^ mo- 
tions and masses, &c, of the heavenly bodies. 

The unit or measure of force of gravity is assumed to be the velocity a falling 
body has obtained at the end of the first second it falls ; this unit is commonly 
denoted by the letter g\ its value at the level of the sea in New York is 
g = 32-166 feet per second, in vacuum. The space fallen through in the first 
second is $g = 16*083 feet. 

This value augments with the latitude, and abates with the elevation above 
the level of the sea. 

I = latitude, h = height in feet above the level of the sea, and r = radius of 
the earth in feet, at the given latitude I. 

r = 20887510(1+0*00164 cos.2Z), 

g = 32-16954(1 — 0-00284 cos.2Z)(l — — . ) 

Letters denote. 
S = the space in feet, which the falling body passes through in the time T. 
u = the space in feet, which the body falls in the Tth second. 
V= velocity in feet per second, of the falling body at the end of the time T. 
T ==■ time in seconds the body is falling. 



VST 



1 2 1 1" 



4 2" 



5 6 9 3" 



8 16 4" 



9 10 25 5" 



11 12 36 6" 



1 


\ 


\ 


221 


\ 


\ 


\ 


\ 




\ 


\ 


\ 


\ 


\ 




V 


\ 




\ 


\ 


\ 

\ 



The accompanying Diagram is a good il- 
lustration of the acceleration of a falling 
body. The body is supposed to fall from a 
to 6, every small triangle represents the 
space 16*08 feet which the body falls in 
the first second ; when the body has reached 
the line 3" seconds, it will be found that it 
has passed 9 triangles, and 9X16*08 = 144*72 
feet the space which a body will fall in 3" 
seconds. The number of triangles between 
each line is the space u which the body has 
fallen in that second. Between 2>" and 4" 
are 7 triangles and 7X16*08 = 112*56 feet, 
the space fallen through in the fourth sec- 
ond. Under the line 3" will be found 6 tri- 
angles, which represents the velocity Fthe 
body has obtained at the end of the third 
second or 6X16*08 = 96*48 feet per second. 
For every successive second the body will 
gain two triangles or 2X16*08 = 32*16 feet 
per second. 



Gravitation. 



195 



Formulas for Accelerated Motion* 



2 S ,— - 



2 






Fa 

"2T 



64-33 ' 






rs 

4-01' 



w = ^-i), 



*-■ =-* 



Example 1. A body is dropped at a height of 98 feet. What velocity will it 
have when it reaches the ground, and what time will it take to fall down ? 

Formula 1. 
Formula 3. 



F = 8-02 ys = 8-02 >/98"= 79-39 feet per second. 

Vs. 



4-01 4<01 

Example 2. A body was dropped at the opening of a hole in a rock, and 
reached the bottom after 3*5 seconds. How deep was the hole? 



Formula 2. 



gT* 



S = ^- = 



■ = 196-98 feet. 



32-16X3-5a 
2 2 

Retarded Motion* 

A body thrown up vertically will obtain inversely the same motion as when 
it falls down, because it is the same force that acts upon it, and causes retarded 
motion when it ascends, and accelerated motion when it descends. 

V== the velocity at which the body starts to ascend. 

v = velocity at the end of the line t. 

T = time in seconds in which the body will ascend. 

t = any time less than T. 

S = height in feet to which the body will ascend. 

s = the space it ascends in the time t. 

Formulas for Retarded Motion* 



t>= V- 



7t 



„ gP> tit* 



■H#l- 7 -r%, 



C = 



- — — — /Z?_ !L? 
" g V 7 2 g ' 



Formulas for I 7 and $ is the same as for accelerated motion. 

Example 3. A ball starts to ascend with a velocity of 135 feet per second. 
At what velocity will it strike an object 60 feet above I Find the time t, by the 
Formula 8. 



135 __ / l35« 2X60 

_ 32-16 \/ 32-162 32-16 

Formula 5, we have, 

v = 135 — 32-16X0-41 



: 0-41 seconds, until it strikes, and from 
= 121-83 feet, per second. 



196 Gravitation. 



Example 4. A ball thrown up vertically from a cannon, occupied 9 seconds, 
until it arrived at the same place it started from. How high up was the ball, 
and at what velocity did it start ? 

One half of 9 = 4| seconds. Formula 2. 

s= sai?>(*e. = 326feethigh . 

F = 32-16X4"5 = 144-7 feet per second. 

If a cannon ball be shot from A, in the direction AB, at an angle BAG to the 
horizon, there are two forces acting on the ball at the same time, namely, — the 
force of gunpowder, which would propel the ball uniformly in the direction AB, 
and the force of gravity which only acts to draw the ball down at an accelerated 
motion ; these two different (uniform and accelerated) motions will cause the 
ball to move in a curved line, (Parabola) AaC. Fig. 225. 

V= velocity of the ball at A. W= weight of the ball in pounds. 

S = the greatest bight of ball over the horizontal line AC. 

t == time from A to C, via a. p = pounds of powder in the charge. 

b — the distance from A to C, called horizontal range. 

/~~p TP~F a p 

V = 2800^ — JT ' P = 78 Jqqqq' 6=243781 sin. x cos. x -j=- 

Example 5. The cannon being loaded sufficiently to give the ball a velocity of 
900 feet per second, the angle x = 45°. Required the distance b = ? and the 
time t = ? 

900*Xsin.45°Xeos.45° - rt .. A , .,,,., „ , . „ 

b = <t2*lfi = 59 feet * * e dlstance fr° m ^ to G - 

It will be observed that the distance b will be longest when the angle x is 45°, 
because the product of sine and cosine is greatest for that angle. sin.45°Xcos. 
45° = 0-5. 

Example 5. What time will it take for a ball to roll 38 feet on an inclined 
plane angle, x = 12° 20', and what velocity has it at 38 feet from the starting 
point. 



2X38 



: 3 k 33 seconds. 



M6Xsin.l2° 20' 
F= g Tsm.x = 3216X3'33Xsin-12° 20* = 22-8 feet per second. 

Power Concentrated in Moving Bodies* 

It is highly important to distinguish betweeu power simply, and power when 
concentrated in a moving body. The former is the force multiplied by its velo- 
city, — but the power concentrated in a moving body is equal to the weight of the 
body multiplied by the square of its velocity, and the product divided by the 
accelleratrix g, — or the power concentrated in a moving body is equal to the 
power expended in giving it the motion. 

Example* A sledge weighing 20 pounds, strikes a nail with a velocity of 12 
feet per second. With what effect did it strike ? 

p== ^W = 89 ' 55 effects * 



Force op Gravity. 



197 





222. 

V = g T sin.* = s/2g S sin.*, 

2 sin.* 2g sin.*' 

g sm.a; v g- sin.* 



223. 

A body will fall from o the distances a, 6, c, 
and <2, in equal times. 



T = 



/25 

vr 




224. 

A body will fall from a to b via c in the short- 
est time, if the curve is a Cycloid. 
S=4d, the length of the Cycloid. 



> n \Ug 



= n> 



2:tg 




225. y 2 sina . cos x 
o=* . 



T = 



V sin.* 



V s in.»* 
"2- ' 




226. 



S = g 



~2M 



V 2 M 
2gF' 



F 



T = 



F = 



V M 
VM 

TV 




2SM 
gT*' 



M=P+Q, and F = P — Q. 



19S 



Centrifugal Force. 



CENTRIFUGAL FORCE. 

Central Forces are of two kinds, centrifugal and centripetal. 

Centrifugal Force is the tendency which a_ revolving body has to 
depart from its centre of motion. 

Centripetal Force is that by which a revolving body is attracted or at- 
tached to its centre of motion. 

The Centrifugal and Centripetal forces are opposites to each other, and when 
equal the body revolves in a circle ; but when they differ the body will revolve 
in other curved lines, as the Ellipse, the Parabola, &c, according to the nature 
of the difference in the forces. If the centrifugal force is o while the other is 
acting, the body will move straight to the centre of motion ; and if the centripe- 
tal force is o while the other is acting, the body will depart from the circle in a 
straight line, tangent to the circle in the point where the centripetal force ceased 
to act. The central forces are distinct from the force that has set the body in 
motion. 

If the centrifugal force be made use of to produce an effect, such effect will be 
at the expense of the one producing the rotary motion. 

Letters denote. 

F '= Centrifugal force in pounds. 
M = the Mass or weight of the revolving body in pounds. 

v = Velocity of the revolving body, in feet per second. 
It = Radii of the circle in which the body revolves, in feet. 

n = number of revolutions per minute. 

Example 1. Required the centrifugal force of a body weighing 63 pounds, and 
making 163 revolutions per minute, in a circle of 4 feet, 4 inches radius ? 



MRn* 63X4-33X1633 



2933 



2933 



= 2475 pounds. 



Example 2. A Railroad train runs 43 miles per hour on a curved track of 
115 feet radii. What should be the obliquity of the track ? 



tan.a; = 



Miles 3 



43» 
69X115 = * 233, 



or x =s 13° 10', the obliquity of the track. 
Example 3. A governor having its arms 1 = 1 foot, 6 inches, how many revol- 
utions must it make per minute to form an angle x = 30° ? 

n = - 7 _= ■ = 47*5 revolutions per minute. 

]/l-5Xcos.30° 




227. 
F= 



r- 



^R 32TM' 

4MMV M Rn* 
60 a £- " 2933 J 



FgR %mF 



J2 = 



Mv ^ 2933F 

F ff - M n* ' 

2933~F 



/2933F _ fFR^ , 



Centritugm, Force Governors. 



199 




228. 



Centrifugal force of a ring. 

M tta s/ Rz+ r * 



F = 



4150 




229. 

Centrifugal force of a grinding stone, 

circle-plane, cylinder, rotating round 

its centre. 



F = 



MRn* 

4150 ' 




230. 

Centrifugal force of a cylinder rotating 
round the diameter of its base. 



p- Mn* \/4 l a 4-3r 
10260 



1231. 




Centrifugal force of a hall, 
(centre of gyration included.) 

Mm \TR*+W r 



2933 




232. 



60 tg 

2933 



h = 



1 = 



Governor. 
54-16 5446 
\/h V J cos. a? ' 

2933 h 

r? cos. a? cos.a?' 






$00 Pendulum. 



PENDULUM. 

Simple Pendulum is a material point under the action of gravitation, 
and suspended at a fixed point by a line of no weight. 

Compound Pendulum is a suspended rod and body of sensible mag- 
nitude, fixed as the simple pendulum. 

Centre of Oscillation is a point in which if all the matter in the com- 
pound pendulum were there collected, it would make a simple pendulum oscil- 
late at the same times. 

Angle of Oscillation is the space a pendulum describes when in mo- 
tion. 

The velocity of an oscillating body through the vertical position, is equal to 
the velocity a body would obtain by falling vertically the distaoce versed sine of 
half the angle of oscillation. 

Letters denote. 

I = length of the simple pendulum, or the distance between the centre of sus- 
pension, and centre of oscillation in inches. 

t = time in seconds for n oscillations. 

n = number of single oscillations in the time t. 

Example 1. Required the length of a pendulum that will vibrate seconds? 
here n = 1, and t = V, 
p. 
I = 39-109 — = 39*109 inches, the length of a pendulum for seconds. 

Example 2. Require the length of a pendulum that will make 180 vibrations 
per minute ? here t = 60" and n = 180. 

, 39\L09£» 39109X60* . OAa . , 
I = -^r - — i8oa— - 4 ' 346 ™***- 

Example 3. How many vibrations will a pendulum of 25 inches length make 
in 8 seconds? 

6-254« 6-254X 8 _ A ., ,. 
n = — ~~ = "~ — =■ = 1° vibrations. 
VI l/25 

Example 4. A pendulum is 137*67 inches long and makes 8 vibrations in 15 
seconds. Required the unit or accelleratrix g = ? 



9- 



. 0-8225Z n* _ 0*8225X137-67X8* = 32 . 209 
fa 15* 



Example 5. A compound pendulum of two iron balls P and Q, having the 
centre of suspension between themselves : see Fig. 238. P = 38 pounds, Q = 12 
pounds, a = 25 inches, and b = 18 inches. How long is the simple pendulum, 
and how many vibrations will the pendulum make in 10 seconds ? 

_ £ P-|_ e = 2i x^x i2 = u . 68inche3 . 

the length of the single pendulum. 

6-254£ 6-254X10 „ AlfAO .,_ A . . .,„ 

n = — 7=r = - . ■ ' = 10-193 vibrations m 10 seconds. 

y/T j/37-68 

If a compound pendulum is hung up at its centre of oscillation, the former 
centre of suspension will be the centre of oscillation, and the pendulum will 
oscillate the same time. 



TENmiLCM AND CENTRE OF OSCILLATION 



201 



/ 
/ 


f~ 

\ 


/ 
/ 

J 
/ 

J 

/ 

/ 
/ 

/ 


I 
\ 
I 

A 


1 

1 
i 

» 

&~4 





233. 

Simple Pendulum. 

1 12g t* S9-lt* 



Tp n* 



t = 



nVl 
6.25' 

0254* 



«/ 




236. 

I TV 1 71% 

0-8225/ 72 2 

o = centre of suspen- 
sion. 

2r 



I = a+ 



5a 



&-, 



■A ^ 



234. 

J. = centre of grav 
ity. 

5 = centre of gyra- 
tion. 

C = centre of oscil- 
lation. 
a: 5 = b : l, 

b= V al^ 1 -1432a, 

I = Ha. 




235. 



Compound Pendu- 
lum. 

r = radius of cylin- 
der. 

16a»+3r a 
l = 12a ' 

7 4a r* 




237. 



, a 2 P-f fr> Q 
■a^PTFQ" 

P and Q expressed 
in pounds, or cubic 
contents. 




JL 



238. 



aP — ^Q 
37=1 P+Q ' 

2 - ^^ Q 
~~^[P+Q)~- 



Length of a Pendulum vibrating seconds at the level of 
At the Equator, lat. 0° 0' 0" 

" Washington, lat. 38° 53' 23" ... - 

" New York, lat. 40° 42' 40" - 

" London, lat. 51° 31' 

" lat. 45° 

« Stockholm, lat. 59° 21' 30" 

I = 39*127 — 0-09982 cos.2 lat. for 



tJie sea, in various places, 
39-0152 inches. 
39-0958 " 
39-1017 " 
39-1393 " 
39-1270 " 
- 39-1845 « 



202 Centre op Gyration. 



CENTRE OF GYRATION. 

Centre of Gyration is a point in revolving bodies in which, if all the re- 
volving matters were there contained, it would obtain equal angular velocity 
from, and sustain equal resistance to, the force that gives it a rotary motion. 

The centre of gyration in different bodies will be found by the accompanying 
formulas, in which x = distance from the centre of motion to the centre of gyra- 
tion. 

Example 1. Pig. 239. Find the centre of gyration in a bar, rotating round one 
of its ends; its length is 7 feet, 3 inches? 

x = 0-5775X7*25 = 413 feet, from the centre of motion. 

Example 2. Tig. 245. Find the centre of gyration of a cone, rotating round its 
vertex, its height being h = 3*3 feet, and H — 8 inches = 0*666 feet. 

* = A /™E^= k /l2X3*3»+3X 0*66~6» = 9 „ fiQ w 
\/ 20 \ / 20 

from the centre of motion. 

Example 3. Fig. 249. A ring or fly wheel having its outer radius R =« 6 feet 
4 inches, the inner radius r = 5 feet 8 inches. Required its centre of gyration 
x = 1 from the centre of motion. 

2 /ft.oo« i (r.cc«. 

' «= 6 feet 

SB 

CONCLUSIONS. 

The object of finding the centre of gyration of revolving bodies is to ascertain 
what effect is necessary to give a mass a certain angular velocity ; or how much 
effect is concentrated in a body having a certain angular velocity. 

Angular velocity is the number of revolutions a body makes in a unit of time , 
it is herein denoted by the letter n. 

Letters denote. 
p = power in effects. 
£T= horse-power. 

F = the Force which is applied to rotate a body, in pounds. 
s = the radius on which the force acts, in feet. 
M = Mass of the revolving body, in pounds. 

x = the distance from the centime of motion to centre of gyration, in feet. 
T = time the force F is applied in seconds. 
iV*= number of revolutions in the time T. 

n = angular velocity or number of revolutions per minute, at tho end of the 
time T. 
g = 32-166 acceUeratrix of the force of gravity. 
G=acceUeratrix of the force F, then, 

a F8* 
G:g = F8*:Mz*, or £ = ^5. 

Example 4. Fig. 249. In connection with the preceding example (3) the fly-wheel 
weighs 7400 pounds. What force i^must be applied at the radius r = 2 feet, to 
give the fly wheel an angular velocity ofn = 128 revolutions per minute, at the 
end of the time T = 40 seconds ? 

nMx* 128X7400X6* mrWO 
Formula 6. F = 153-5^ is = 153.5x40x2 "" = ■ P ound8< 

How many revolutions did the wheel make in the 40 seconds ? 

, 2-56T3 FS 2-56X40*2X2773X2 OK ntr . ,. M 
FormuU 9. N= -j^ - 7400x6* " 85 * 27 re ™ lutlon9 ' 



Centre op Gyration. 



203 



239. 



A line or bar* 
x = 0.5775/, 
x - 0-2887Z. 




240. 
A circumference round its diameter, 
A circle-plane round its centre, 
A cylinder round its axis. 

x - 0-7071r. 



241. 




A circle-plane round its diameter. 
a = 4 5r. 




242. 



A Sphere round its diameter. 
Convex surface, x =. 0-8165r, 
Solid, . . . « = 0-6324r. 



S 



-3t- 



Nl 



3 



243. 



Parallelopiped. 



VT2- 



4ft+fr 
12 



+a a +a J. 




244. 



Cylinder. 



'-\HF. 



204 



Centre of Gyration. 




245. 






Cone. 
2A»+3i?» 
20 * 

12/i 2 +3.R 2 



20 




246. 



Conic Frustrurn. 



*Vb( 



R*-tSRr+Rr* \ 
10 v R*+Rr+r* ' + 



3 ,22- 



20\R~^r~ a J 



247. 



Cylinder and Sphere, 






a?= Va a +£r% 



* = VoH|f 



* 



248. 




Wedge and Ring, 
x = 0-204 V12Z 2 +^ a +Z> 3 , 



249. 




F/r/ "Wheel. 
F G : AT # = a? a : s\ 




250. J7y WfoeZ tw'M Arms. 
a? a (M+?w) = JVf — o— +wi 12 

/6M(fi Q +r g )+m(4r»+fr > ) 
* V l"2(M+m) 



Centre of Gyration. 



205 



Formulas for Force and Power of Acceleration* 

2-56T a J Fs 



m f\n s N 



T -*\f' 



4tzNM 
Fs 



, 2, 



T = 



2rt s n 




2-56 T a 5 ' 



- 3, 

- 4, 

- 5, 



nMx* 
r ^itottt:, - - 6, 



153-5 T s 

N Mx* 
2-56 T*F' 

n M x* 
153-5 TF' 



, - - 7, 



iV = 



n = 



M = 



T = 



M,ra 


•*» 


153-5 T Ps 


-10, 


M x* ' 


n* Mx* 


•11, 


244 T ' 


244 TP 

n* x* ' " 


■12, 


244P ' 


-13, 


/244TP 


- 14. 



lMTOOT" " ' 

jy- uaoorg 16f 



Fly-Wheels. Weight of. 

The weight of a fly-wheel will be determined by the formula 16 in which the 
time T = 130 seconds, the time in which the fly-wheel would concentrate the 
game power as the steam-engine. When the works or resistance is very irre- 
gular it will be better to take the time T = 170. The centre of gyration (in- 
cluding ring and arms,) can in practice be assumed at x = r the inner radius of 
the ring. 

Example 5. Required the weight of a fly-wheel for ordinary work, the steam 
engine being 56 horse power, making 42 revolutions per minute, and the inner 
radius r = 10 feet ? 

__ 1 34100 T K 134100X130X56 ecoc 

*= — ^ " = 42*X10» = 5535 V ° UndS - 



206 



Centre of Gravity. 



CENTRE OF PERCUSSION. 

Centre of Percussion is a point in which the momentums of a moving 
body are concentrated. Centre of Percussion is the same as centre of oscillation, 
and to be calculated by the same formulas. 

Take an iron bar in one hand, and strike heavily over a sharp edge, if the 
centre of percussion of the bar strikes over the edge, the whole momentum will 
there be discharged, but if it strikes at a distance from the centre of percussion a 
part of the momentum will be discharged in the hand, and a shock felt. 

It is sometimes of great importance to properly place the centre of percussion. 
If it is dislocated, the moving body not only fails to properly transmit its effect, 
but the lost momentum acts to wear out the machinery. 



CENTRE OF GRAVITY. 

Centre of Gravity is a point around which the momentums of all matters 
(under the action of the force of gravity) in a body, or system of bodies, are 
equally divided. 

A body or system of bodies suspended at its centre of 
gravity, will be in equilibrium in all positions. 

A body or system of bodies, suspended in a point out of 
its centre of gravity, will hang with its centre of gravity ver- 
tical under the point of suspension. 

A body or system of bodies suspended in a point out of 
its centre of gravity, and having two different positions, 
the two vertical lines through the point of suspension 
will meet in the centre of gravity ; thus if a plane be hung 
up in two different positions, the vertical lines a, 6, and 
c, d, will meet in the centre of gravity o. 

z = distance to the centre of gravity as noted in the 
figures. 

„/ Examplel. The radius of a circle being 3 feet, how far is 
its centre of gravity from the centre of the half circle ? 
z = 0-6367X3 = 1'91 feet. 

Example 2. How far from the bottom of a cylindric shell, 
open at one end, is its centre of gravity 1 The cylinder is 
4 feet long, radius r = 0-8 feet. 

X. A 

= 0*625 feet. 




r+2/i 0-8+2X4 

Example 3. Fig. 264. An irregular figure weighing P = 138 pounds, is sus- 
pended between a fulcrum and a weight, I = 5*6 feet, W= 57 pounds. Re- 
quired the distance to the centre of gravity z = ? 



138 



Centre of Gravity. 



207 




Quadrangle. — a and b parallel. 
z ~2 S^TuT)' 




253. 



Triangle, 
h 




254. 
Half a circle plane or Elliptic plane. 
z = 0-424r. 




255. 



Circle sector. 
2c r 
Z =to- 





256. 



257. 



Circle Segment, a = area. 

* ~ 12a 

a? = A+2 — r. 

Parabola. 
2h 



For half a Parabola x = rrb. 

o 



208 



Centre op Gravity. 




7- » 



258. 



Half Sphere. 



Convex* surface 
Solid .... 







259. 



Solid, 



Spherical Sector. 



«-i (--*-)• 








260. 

Convex surface 5? 



Solid 



Spherical Segment. 
h_ 
2' 

h r2r*+K 



2'l_3r*+A*'J 



261. 



Cone. 



Convex surface 2 = ^-, 



Solid 




262. Conic Fustrum. 

h hTR-'i 
Con. sur. z 



Solid 



z > 



h h T R — r "I 
2 6 L #+r 'J 



* rft a +r(2i?+3r) 1 



263. 




Fyramidic Fustrum. 



A and a = area of the two bases. 
h fA+Za+2y/Aa 1 



Solid 



4 L A+a+>/-A a 



Centre op Gravity. 



209 




264. 



Irregular Figure. 

P: W=l:z. 
Wl 




(L 



1 — *-£■ 




< & 





265. 

To find the Centre of Gravity of two 
Bodies, P and Q. 

Qa . Pa 



P-tQ 9 P+Q* 



266. 

To ,/Jwc? the Centre of Gravity of a sys- 
tern of bodies. 



6 = 



Ra 
P+R' *~P-i-R+Q' 



Qd 



267. 

Half a circumference of a Circle or 

Ellipse. 

z = 0-6367r. 





268. 



Circle arc or Elliptic arc. 

c r c(c»+4A») 

z " b Mb ' 



269. 

Cylindric Surface with a bottom in one 
end. 



* = 



r+2A' 



18* 



21u 



Specific Gravity. 



SPECIFIC GRAVITY. 

Specific Gravity is the comparative density of substances. The unit for 
measuring the specific gravity is assumed to he the density of rain water, or 
distilled water. 

One cubic foot of distilled water weighs 1000 ounces, or 62*5 pounds avoir- 
dupois. 

To Find tlie Weight of a Body. 

RULE 1. Multiply the contents of the body in cu»bic feet by 62*5, and the 
product by its specific gravity, will be the weight of the body in pounds 
avoirdupois. 

RULE 2. Multiply the contents of the body in cubic inches by 0*03616, 
and the product by its specific gravity, will be the weight of the body in 
po ands avoirdupois. 

RULE 3. Divide the specific gravity by 0*016 and the quotient is the weight 
of a cubic foot. 

Example 1. A bottle full of mercury is 3 inches, inside diameter, and 6 inches 
high. How much mercury is there in the bottle in pounds? 

One cubic inch of mercury weighs 0*491 pounds, and by the formula for 
Fig. 119 we have the 

weight = 0*491X0*785X32X6 = 20*85 pounds. 

Example 2. Required the weight of a cone of cast iron, diameter at the 
base d = 1*33 feet, height h = 4 feet? One cubic foot of cast iron weighs 
450*5 pounds, and by formula for Fig. 117 we have the 

weight = 450*5X0*2616Xl'332X4 = 834 pounds. 

Example 3. The section area of the lower hole in a steam boat is 245 square 
feet ; how much space must be taken in the length of the hole for 131 tons 
of anthracite coal ? 

Anthracite coal are 42*3 cubic feet per ton.; 



length = 



42*3X131 
: 245 



: 22*6 feet, the space required. 



Weight and Bulk of Substances* 



Names of Substances. 



Cast iron, 
Wrought iron, 
Steel, 
Copper, - 
Lead, 
Brass, 
Tin, - 
Piue, white 

" yellow, - 
Mahogany, 
Marble, common, 
Mill-stone, 
Oak, live - 

" white, - 
Clay, 

Cotton Bales, • 
Brick, 
Plaster Paris, - 



Cubic 


Cubic* 


feet 


foot 


in 


per. 


pounds. 


ton. 


450*5 


4*97 


486*6 


4*60 


•489*8 


4*57 


555- 


4-03 


707*7 


3*16 


537*7 


4-16 


456 


4-91 


29*56 


75-6 


83-81 


66*2 


66*4 


33-8 


141*0 


15-9 


130 


17-2 


70 


32*0 


45.2 


49-5 


101*3 


22-1 


100 


22*4 


105 


21*3 



Names of Substances. 

Sand, 
Granite, - 
Earth, loose, - 
Water, salt, (sea) 

" fresh - 
Ice, - 
Gold, 
Silver, 
Coal, Anthracite 

" Bituminous 

" Cumberland 

" Charcoal 
Coke, Midlothian 

" Cumberland 

" Natural Virginia 

Conventional rate of 
Stone coal, 28 bushels 
(5 pecks) = 1 ton, 



Cubic 

feet 
in 
pounds 
94*5 
139 
78*6 
64*3 
62*5 
58*08 
1013 
551 

53 

50 

53 
18*2 
32-70 
31*57 
46*64 



Cubic 
foot 
per 
ton. 

23-7 
16*1 

28*5 

34*8 

359 

38*56 

2-21 

4*07 

42*3 

44*8 

42-3 

123 

68*5 

70*9 

48*3 



43*56 



Specific Gravity. 211 



To Find the Specific Gravity. 

W= weight o-f a body in the air. 

w = weight of the body (heavier than water) immersed in water. 

8 = specific gravity of the body. Then, 

W—w. W=1:S. S= „ r W , - --- 1, 

Example 4. Required the specific gravity of a piece of iron-ore weighing 
6-345 pounds in the air, and 4-935 pounds in water, S = t 

fl.OlS 

-, = 4*5 the specific gravity. 



6-345 — 4-935 

When the body is lighter than water, annex to it a heavier body that is ablo 
to sink the lighter one. 

S = specific gravity of the heavier annexed body. 
s = specific gravity of the lighter body. 
W= weight of the two bodies in air. 
w = weight of the two bodies in water. 
F= weight of the heavier body in air. 
v = weight of the lighter body in air. 



IF-t*-- 



Example 5. To a piece of wood, which weighs t> «= 14 pounds in the air, is 
annexed a piece of cast-iron V= 28 pounds; the two bodies together weigh 
w = 11-7 pounds in water. Required the specific gravity of the wood? 
W= V+v = 28+14 = 42 pounds. 
8 = 7*2 specific gravity of cast-iron. 

14 
Formula 2. S= ^ = 0-529, the specific 

42-UT-l? 

7-2 

gravity of the wood, (Poplar White Spanish.) 

A simple way to obtain the specific gravity of woods, is to form it to a parallel 
rod, and place it vertically in water, then when in equilibrium, the immersed 
end is to the whole rod as the specific gravity is to 1. 

Example 6. A cylinder of wood is 6 feet, 3 inches long, when Immersed verti- 
cally in water it will sink 3 feet, 9 inches by its own weight. Eequired its spe- 
cific gravity. 

3-75 : 6-25 = S : 1, S = 4? = 0-600. 
1 6-25 

To discover the Adulteration in Metals, or to find the proportions of two Ingredients 
in a Compound. 
v _ W-s{W-w) g 



Example 7. A metal compounded of silver and gold weighs W= 6 pounds 
in the air, and in water w = 6'636 pounds. Require the proportions of silver 
and gold 1 

S= 19-36 specific gravity of gold. 
s = 10-51 specific gravity of silver. 

. , . _ 6 — 10.51(6 — 5.636) . w „ . _ ._ 

weight V= ^ f = 4-755 pounds of gold. 

t 10-51 

19-36 an< * 1*245 pounds of silver. 



212 



Specific Gravity. 



Names of Substances. 

Metals* 

Platinum, rolled - - 
" wire, - - 

" hammered, 

" purified, 

" crude, grains 

Gold, hammered - - 
" pure cast - - - 
" 22 carats fine - 
« 20 " " - 
Mercury, solid at — 40° 
" at +32° Fahr. 

" « 60° 

« « 212° 

Lead, pure - - - 
" hammered - 
Silver, hammered - 
" pure - - - 
Bismuth, - - - - 
Red Lead, - - - - 
Cinober, - - - - 



Copper, wire and rolled 

" pure - - - - 

Bronze, gun metal - 

Brass, common - - - 

Steel, cast steel - - - 

" common soft - 

" hardened & temp. 

Iron, pure 

" wrought and rolled 

" hammered 

" cast-iron - 

Tin, from Bohmen 

" English - 
Zinc, rolled - 
" cast - - 
Antimony, - 
Aluminium • 
Arsenic, - - 
Stones and Earths* 
Topaz, oriental 
Emery, - - - - - 
Diamond, - - - - 
Limestone, green - 
" white - 

Asbestos, starry - 
Glass, flint - - - 
" white - - - 
" bottle - - - 
" green - - - 
Marble, Parian - - 
" African - 
" Egyptian - 
Mica, - - - - - 
Hone, white razor 

Chalk, 

Porphyry, - - - - 

Spar, green - - - 

" blue - - - 



! 


Weiglu 


Specific 


per 


gravity. 


cubic 




inch. 


22-669 


•798 


21-042 


•761 


20-337 


•736 


19-50 


•706 


15-602 


•565 


19-361 


•700 


19-258 


•697 


17-486 


•733 


15-702 


•568 


15-632 


•566 


13-619 


•493 


13*580 


•491 


13-375 


•484 


11-330 


•410 


11-388 


•412 


10-511 


•381 


10*474 


•379 


9-823 


•355 


8-940 


•324 


8-098 


•293 


8-030 


•290 


8-878 


•321 


8*788 


•318 


8-700 


•315 


7-820 


•282 


7-919 


•286 


7*833 


•283 


7-818 


•283 


7-768 


•281 


7-780 


•282 


7-789 


•282 


7-207 


•261 


7-312 


•265 


7-291 


•264 


7-191 


•260 


6-861 


•248 


6-712 


•244 


2'5 


0-09 


5-763 


•208 


4-011 


•145 


4-000 


•144 


3-521 


•127 


3-180 


•115 


3-156 


•114 


3-073 


•111 


2-933 


•106 


2-892 


•104 


2-732 


0987 


2-642 


0954 


2-838 


103 


2-708 


0978 


2-668 


0964 


2-S00 


1000 


2-838 


104 


2-784 


100 


2-765 


0999 


2-704 


0976 


2-693 


•0971 



Names of Substances. 



Alabaster, white • 
yellow - 
Coral, red - - - - 
Granite, Susquehanna 
Quincy - 
Patapsco - 
Scotch - - 
Marble, white Italian 

" common - 
Tale, black - 
Quartz, - - - 
Slate, - - - 
Pearl, oriental 
Shale, - - - 
Flint, white - 

black • 
Stone, common 

Bristol 

Mill - 

Paving 
Gypsum, opaque 
Grindstone, - 
Salt, common 
Saltpetre, - - 
Sulphur, native 
Common soil, 
Rotten stone, 
Clay, - - ■ 
Brick, - - • 
Nitre, - - • 



Plaster Paris, 

Ivory, - - 

Sand, - - 

Phosphorus, 

Borax, - - 

Coal, Anthracite - 

Maryland - - - 

Scotch - - - - 

New Castle - - 

Bituminous - - 

Charcoal, triturated - 

Earth, loose - - - - 

Amber, ------ 

Pimstone, - - - - 

Lime, quick - - - - 

Charcoal, - - - - - 
Woods (Dry*) 

Alder, - - - 

Apple-tree, - 

Ash, the trunk 

Bay-tree, - - 

Beech, - - - 

Box, French - 
" Dutch - 
Brazilian red - 

Cedar, wild - - - 
Palestine - 



I Weight 
Specific 1 per 
gravity J cubic 
inch. 



2-730 
2-699 
2-700 
2-704 
2-052 
2-640 
2-625 
2*708 
2-686 
2-900 
2-660 
2-672 
2-650 
2-600 
2-594 
2-582 
2-520 
2-510 
2-484 
2-416 
2-168 
2-143 
2-130 
2090 
2-033 
1-984 
1-981 
1-930 
1-900 
1-900 
1-872 
2-473 
1-822 
1-800 
1-770 
1-714 
1-640 
1-436 
1-355 
1-300 
1-270 
1-270 
1-380 
1-500 
1-078 
1-647 
0-804 
0-441 



•793 

•845 
•822 
•852 
•912 
1-328 
1-031 
•596 
•613 



Specific Gravity. 



213 



Names of Substances. 



Cedar, Indian - 

" American 
Citron, 
Cocoa-wood, 
Cherry-tree, • 
Cork, 

Cypress, Spanish 
Ebony, American 

" Indian 
Elder-tree, 
Elm, trunk of - 
Filbert-tree, - 
Fir, male - 
" female 
Hazel, 

Jasmine, Spanish 
Juniper-tree, - 
Lemon-tree, . - 
Lignum-vitse, - 
Linden-tree, - 
Log-wood, 
Mastic-tree 
Mahogany, 
Maple, 
Medlar, - 
Mulberry 
Oak, heart of, 60 old 
Orange-tree, - 
Pear-tree, 
Pomegranate-tree, 
Poplar, - 

" white Spanish 
Plum-tree, 
Quince-tree, - 
Sassafras, 
Sprure, - 

" old 
Pine, yellow - 

" white 
Vine, 
Walnut, - 
Yew, Dutch 

" Spanish - 
Liquids. 
Acid, Acetic - 

" Nitric - 

" Sulphuric 

'" Muriatic 

" Fluoric - 

" Phosphoric 
Alchohol, commercial 

" pure 
Ammoniac, liquid 
Beer, lager 
Champagne, - 
Cider, 

Ether, sulphuric 
Egg, - - - 
Honey, 

Human blood, 
Milk, 





Weight 


Specific 


per 


gravity. 


cubic 




inch. 


1-315 


•0476 


•561 


•0203 


•726 


•0263 


1-040 


•0376 


•715 


•0259 


•240 


•0087 


•644 


•0233 


1-331 


•0481 


1-209 


•0437 


•695 


•0252 


•671 


•0243 


•600 


•0217 


•550 


•0199 


•498 


•0180 


•600 


•0217 


•770 


•0279 


•556 


•0201 


•703 


•0254 


1-333 


•0482 


•604 


•0219 


•913 


•0331 


•849 


•0307 


1-063 


•0385 


•750 


•0271 


•944 


•0342 


•897 


•0324 


1-170 


•0423 


•705 


.0255 


•661 


•0239 


1-354 


•0490 


•383 


•0138 


•529 


•0191 


•785 


•0284 


•705 


•0255 


•482 


•0174 


•500 


•0181 


•460 


•0166 


•660 


•0239 


•554 


•0200 


1-327 


•0480 


•671 


•0243 


•788 


•0285 


•807 


•0292 


1-062 


•0384 


1-217 


•0440 


1-841 


•0666 


1-200 


•0434 


1-500 


•0542 


1-558 


•0563 


•833 


•0301 


•792 


•0287 


•897 


•0324 


1-034 


•0374 


9-97 


•0360 


1-018 


•0361 


•739 


•0267 


1-090 


•0394 


1-450 


•0524 


1-054 


•0381 


1-032 


•0373 



Names of Substances. 



Oil, Linseed - 
" Olive - 
Turpentine 
Whale 
Proof Spirit, ~ - 
Yinegar, - 
Water, distilled 
Sea 

Dead sea 
Wine, 
' Port 



Miscellaneous. 



Asphaltum, 

Beeswax, - 

Butter, - 

Camphor, 

India rubber, 

Fat of Beef, 
Hogs, 
Mutton, 

Gamboge, 

Gunpowder, loose 

shaken 

solid 

Gum Arabic, - 
Indigo, - 
Lard, 
Mastic, - 
Spermaceti, - 
Sugar, 
Tallow, sheep - 

calf 

ox, 
Atmospheric air, 



Gases* Vapours. 

Atmospheric air, 
Ammoniacal gas, - 
Carbonic acid, - 
Carbonic oxid, 
Carburetted hydrogen, 
Chlorine, - 

Chlorocarbonous acid, 
Chloroprussic acid, 
Flouboric acid, 
Hydriodic acid, 
Hydrogen, 
Oxygen, - 

Sulphuretted hydrogen, 
Nitrogen, 
Yapour of Alchohol, 

" turpen'e spir., 

" water, 
Smoke of bitumin. coal, 

" wood, 
Steam at 212° - 





Weight 


Specif c 


per 


gravity. 


cubic 




inch. 


•940 


•0340 


•915 


•0331 


•870 


•0314 


•932 


•0337 


•925 


•0334 


1-080 


•0390 


1-000 


•0361 


1-026 


•0371 


1-240 


•0448 


•992 


•0359 


•997 


•0361 


•905 


, *0327 


1-650 


0597 


•965 


'0349 


•942 


# 0341 


•988 


# 0357 


•933 


0338 


•923 


'0334 


•936 


•0338 


•923 


*0334 


1-222 


•0442 


•900 


•0325 


1-000 


•0361 


1-550 


•0561 


1-800 


•0650 


1-452 


•0525 


1-009 


•0365 


•947 


•0343 


1-074 


•0388 


•943 


•0341 


1-605 


•0580 


•924 


•0334 


•934 


•0338 


•923 


•0334 


•0012 


.....43 




Weight 




cub. ft. 




grains. 


1-000 


527-0 


•500 


263-7 


1-527 


805-3 


•972 


512-7 


•972 


512-7 


2-500 


1316 


3-472 


1828 


2-152 


1134 


2-371 


1250 


4-346 


2290 


•069 


36-33 


1-104 


581-8 


1-777 


9370 


•972 


512-0 


1-613 


851-0 


5-013 


2642 


•623 


328-0 


•102 


53-80 


•90 


474-0 


•488 


257-3 



214 



Hydrometer. 




HYDROMETER. 



A eodt wholly immersed in a liquid will lose as much of its weight, as the 
weight of the liquid it displaces. 

A floating body will displace its own weight of the 
liquid in which it floats. 

A cylindrical rod of wood or some light materials, 
being set down in two liquids, A and B, of different 
specific gravities, when in equilibrium it will sink to 
the mark a in the liquid A, and to b in the liquid B; 
then the specific gravity of A : B = 6, c : a, c, or in- 
verse as the immersed part of the rod. This is the 
principle upon which a hydrometer is constructed. 



Table showing the comparative Scales of Guy Lussac and Baumb, with the Specific 
Gravity and Proof, at the temperature of 60° Fahr. 



271. 




Guy Lussac's. 


Buum 


100 


46 


95 


40 


•S 90 


36 


•9 85 


33 


8 80 


31 


* 75 


28 


t 70 


26 


§, 65 


24 


«m 60 


23 


2 55 


21 


$ 50 


19 


-g 45 
2 40 


18 


17 


Z 35 


16 


& 30 


15 


* 25 


14 



Specific Grav, 


Proof. 


•796 


1001 




•815 


92 


u 


•833 


82 


k 


•848 


72 


® . 


•863 


62 




•876 


52 


r-M o 
P 1 i" 


•889 


42 


o> P* 


•901 


32 


ti 


•912 


22 


P* 


•923 


12. 




•933 


Proof. 


•942 


8 ] 


•951 


18 S«g' 


•958 


29 h § 

35 p* 


•964 


•970 


48] 





HYDROSTATICS. 

Letters denote. 

A and a = areas of the pressed surfaces in square feet. 

P and p = hydrostatic pressure in pounds. 

d = depth of the centre of gravity of A or a under the surface of the liquids 
in feet. 

S = specific gravity of the liquid. 

Example 1. Fig. 272. The plane A = 3*3 square feet, at a depth of d = 6 feet 
under the surface of fresh water. Required the pressure P — ? Specific gravity 
of fresh water S «■ 1. 

P= 62-5 A d — 62-5X3-3X6 = 1237-5 pounds. 

Example 2. Fig. 275. The area of the pistons A = 8-5 square feet, a = 02 
square feet, I = 4 feet, e = 9 inches, and F= 18 pounds. Required the pres- 
sure P = ? 



P = 



FIA 18X4X8-5 



= 40800 poundf. 



•e a 0-75X0*02 

It must be distinguished that the centre of pressure and centre of gravity of 
the planes, are two different points ; the centre of pressure is below the centre 
of gravity, when the plane is inclined or vertical. 



Hydrostatics. 



215 




272. 



P = 62-5 SAd 9 
P 



A= 
d = 



62-5 Sd f 

P 
62-5 S A. 




273. 



The Hydrostatic paradox. 



The pressure P is independent of the 
width of column C. 

P = 62-5 & J. A. (same as above.) 




274. 



P = A(62-5Sh+£\ 

, Pa-pA 
62-5 &A a 




P = 



P = 



275. Bramatis Hydraulic Press. 

FIA A- Pea 

TiT' "FT 

Pea n FAl 




276. Centre of Pressure of a rectangle, 
the upper edge at the surface 

of the liquid d = % h. 

277. Centre of Pressure of a triangle, 
the base being at the surface of 

the liquid, d = i A. 




278. Centre of Pressure of a 
triangle, the vertex being at the surface 
of the liquid. d = i A. 

279. 



d = ^ + V4( h-h'Y + h\ 



216 Stability of Vessels in Water. 



STABILITY OF VESSELS IN WATER. 

Letters denote. — D= displacement of the vessel in pounds. $£ = greatest 
immersed section in square feet. B = breadth of beam in feet in the water-line. 
L = length of the vessel in feet in the water-line. a = the vertical distance in 
feet between the centres of gravity of the vessel and displacement when in 
equilibrium. When the centre of gravity of the vessel c is below that of the 
displacement e, as in Fig. 1, then a is positive, or -\-a ; and when c is above e, 
as in Fig. 2, a is negative, or — a. b = horizontal distance in feet from the centre 
of gravity of the displacement when in equilibrium to the same centre when 
out of equilibrium. d = depth of the centre of gravity of the displacement 
under water-line in feet when in equilibrium ; and cT= depth of the same centre 
when out of equilibrium. / = force of wind in pounds per square inch (see 
page 233). h= vertical height in feet from the centre of gravity of the dis- 
placement to the centre of the weight W, Fig. 281, when the vessel is in equi- 
librium. r= horizontal distance in feet from the centre of the vessel to the 
centre of the weight W, Fig. 281. 1 = leverage in feet upon which any force 
acts to careen the vessel, to be calculated from the centre of gravity of the dis- 
placement, perpendicular to the direction of the careening force. In sailing, 
I is taken from the centre of gravity of the displacement to the centre of effort 
of the sails. m = vertical distance in feet from the centre of gravity of the 
displacement when out of equilibrium to the metacentre ra. v = careen angle 
of the vessel. tc = angle of the sails to the length of the vessel. z= angle of 
the wind to the sails. (^ = area of resistance of the vessel in sq. ft. (see page 
271). A = area of all the sails in square feet. 3f= miles or knots per hour, by 
sailing. .F=forcein pouuds acting to propel the vessel forward. If = any 
weight or force in pounds acting on the level I to careen the vessel. 

Example 1. — The U. S. steam frigate Niagara is £ = 329 ft. long; B — 55 ft. 
wide ; greatest immersed section, jgT = 855 sq. ft. ; displacement, D = 11,200,000 
pounds ; vertical distance between the centres of gravity of displacement and 
vessel assumed to be — a = 2.5 ft. What momentum (Wl = ?) is required to 
careen her to an angle of v— 8°, and what force (W= ?) is required on a lever 

of 1 = 35 feet? , , 

55 3 Xtan.8° / 11,200,000 
Formula 1. b = 12X855 \/ 64.3 X 329 X 855 ^ 

The required careen momentum will be 
Formula 2 W 1 = 11,200,000 (1.945—2.5 sin. 8°)= 17,887,520 foot pounds, 
17,887,520 

and the force W=— — = 511,072 pounds = 228 tons. 

35 

Example 2. — rt'is required to find the momentum of stability of a man-of-war, 
by moving a number of guns of known weight from one side to the other. Each 
gun weighs 25,000 pounds, and four guns are moved to the opposite side, to r 
= 20 feet from the centre of the vessel ; the height of the centre of gravity of the 
guns above the centre of gravity of displacement is h= 16 feet. There will be 
eight guns of 25,000 pounds, or IF = 200,000 pounds careen weight on one side, 
by which the vessel is careened to an angle of v = 7° 20'. Dimensions of the 
vessel are D = 6,150,000 pounds, B = 40 feet, L = 260, and g£ = 566 square feet. 
Required the vertical distance between the centres of gravity of the vessel and 

displacement, a = ? , 

40 3 X tan. 7° 20' / 6,150 , 000 __ 

Formula 1. b = 12 x 6 ^ \^ ^ 3 x 260 x 666 - > 057 fe et. 

Formula 6. Z = sin. 7°20'(16 — 20Xsin. 7 o 20'H-20sec. 7020'— ,1.057 =21. 36 ft. 

1 /200,000 V 21.09 \ 

t / , *- ^ J * — --± —1.057 J = — 2.84 feet. 

7°20'V 6,150,000 

a is negative when — — -< 0. 



A=— 2.8 



Stability of Vessels is Water. 




280. 



6 = 



JS a tan.v 3 / J) 



V 64,3* 



217 



12.® V 64.3* Z, ST 
Wl = D (b± a sin. v), . . . 2 



Meta- 

centre 



ra = — - — , \ / _ 

12lV 64.3*. 



IS V U.Z*Lm 




£^rzzr: Depth S = d cos. J v, . 



281. 



1 /TV I 



sin. v\D /' • u 

?=sm.v(A^rsin.i;)+rsec.v~6 J 6 

cot * v ~m( m±a )> • 7 

Capsizes when a sin. v = or > 6, 8 



282. 



Careen force TT=/^ sin. 2 cos. u, 9 
Sailing force i^=/^ sin. z sin. w, 1 



Miles per hour M 



283. 



"~\6~e' ' H 



sinTTsinT 
* 64.3 for salt and 62.5 for fresh water. 



Force of wind/ == - 

A sin. z sin. u 



218 Hydraulics. 



HYDRAULICS. 

^ Let the vessel A, Fig. 284, be kept constantly full of water up to the water 
line w. In two horizontal faces lower than the water line w, are made orifices 
a and a', through which the water will pass up vertical nearly to the water 
line it). Omitting the resistance of air, &c, the jet should theoretically reach 
the water line w ; practically it reaches 0*967ft. 

It is evident that the velocity of the jet through the orifices, must be the ve- 
locity due to a body falling the height h, according to the law of force of 
gravity. 

Letters denote. 

Q = actual quantity of water discharged per second or in the time t f in cubic 
feet. 

h — head, or height of water over the orifice. 
t = operating time in seconds. 
a = area of the orifice in square feet. 
m — the coefficient for contraction. (See Tig. 299 ) 
G = gallon of 231 cubic inches discharged in the time t. 
V= velocity through the orifice in feet per second. 

Example 1. Fig. 284. How many gallons of water will be discharged in five min- 
utes, through an orifice of 0*025 square feet, applied at 8 feet under the level of 
the water? 

G = 37*75a t yh = 37-75X0*025X5X60 j/8 = 800 gallons. 
Fig. 285. The weight P can represent the weight of a column of water whose 

P N 

height = „ , . ■ = A ^,„ , acting on the area A. 
62*0.4 0*96< ' 

Fig.286.n = number of down strokes per minute, s = stroke of piston; the 
air vessel C— QA s at the pressure of the atmosphere. 

Example 2. Fig. 286. How many double strokes must be made per minute by 
the lever of a hand pump, to throw up 22 cubic feet of water 18 feet high, in the 
time of 8 minutes and 15 seconds ; the levers I = 30 inches, e = 8 inches, 
s = 0*6 feet, F = 20 pounds ? 8X60+15 = 495 seconds. 

Z630Q7i'e 3630X22X18X8 e . _ , . . , 

n = - * = in-vn.fivy.vV fl = ^ strokes per minute, 
tsFl 49oX0 OX20X30 * 

Example 3. Fig. 294. A vessel of rectangular form is of dimensions J. = 6 
square feet, the Wight h = 5 feet. What time will it take the water level to 
sink 2 feet, when the orifice a = 0*212 square feet. 

AQi-«) = 6(5-8) = 5 . 66< 

2-b2a(yh+y?v 2*52X0*212(]/5+y/3) 

Motion of Water in Pipes* 

Letters denote. 

L = extreme length of the pipe in feet. 

d = inside diameter in feet, and uniform throughout the length L. 

Example 4. Fig. 287. What will be the velocity of the water through a pipe of 
45 feet inside diameter, and L — 6S feet long, the head pressure of water being 
ft = 8 feet? 



'—.tf, 



, = 9*6 feet per second. 



68+50X0*45 



Hydraulics. 



219 



*;w 



hi 






u 




284. 7=8-02 %/ F" 

Q^mat 8-02 vT= 5-05 a * VF 

G^Zl'll at\Th, m = 0-63, , 
q* /et = 0-967 A , 



A = 



25-5 a 3 1* 



285. 
F= 1-015 



Vr q = ai 



1 i- *-«ra 




286. 



n * K /— > 3630 Q A' c 

Q-«*V **' w= — CTT' 

?=7-5a*\/-ZV A'- ^— • 




287. Motion of Water in Pipes. 

„ ,„ /"&*> / dh 



J = ' 24 \/ IT* 



142 d* ' 



>- 



\F 



288. 



r=e- 



V 



Motion of Water in Pipes. 

d F 



D(L+bQ<d 



III Jl IJJOO* 

VfdF 




289. 



rdiF-^D^i) 



220 



Hydraulics. 




290. 



Weirs. 



Q = kh I. See Table for Weirs* 
f— £ *- Q 





291. 



Q = 5-35m £ A f v^ 
5<35m£Av/A' 



292. 



Q = 5-35m 5 t(hVh — A' vT^" 
G = 40m b t(h</T— h'JT), 

Q 



t = 



5-35m b(hVh — /iWh') ' 




293. 



^ = 0-95m i(y/^- ,/£' ) 

^4 = area of the vessel in square feet. 
t = time in seconds, in which the water level 
will sink the space h — h\ 



294. 



Am a (^/h+^/h'), 
Q = 4mat(y/'h+</ffJ f 



___l_ - 


-a~/t~ 


-\ 


S11SBS 


h 

A 


t 


^=7E=EF& JUS: 











295. 



t=> 



385a 



3-85*' m 



ASh 



3*85am 



IlTDBilTLICS. 



221 




296. 



AAWh 



h = 



13-7ma>/ A + A' 

Ah' , 

A + A' 



^-- — = 


r 




= -= 


===H! 


-'— — 




^^ 



297. Short Drain. 

y _ 8-02 >rr ^ 

»== v / 7 a + 32-l A' , 
Q = a mVt. from Vtov about 6 V a , 




298. 



r = 



Long D rain . 
8-02 VA 



5 = 6+2d 



F a + 64- 32rt'- 0-007 



slV< 



l=Vtov, feet. 



299. Proportions of the contracted Vein. 

a:?n = l0' 2 : 8*. ro = 0-64a. 
m = 0-64 when contracted on 4 sides, 
m = 0-72 " " " 3 sfdes. 

m = 0-8 " " " 2 $t<fc». 

w = 0-9 " " " 1 side. m 



300. 



The form of the Vein is a Parabola, 
d = 2VhT, _V=8 s fnTT' 
Q = 8mat Vh, 



tan v = ?*- 
a 






301. 



x = sin' z A y = stn. 2 cos. * h. 



d = 2 V {h! + x ) (h - x ) -y 



222 



Hydraulics. — Hydrodynamics. 



Example 5. Fig. 289. Required the velocity and quantity of water discharged in 
a long pipe or hose of L = 135 feet long, and d = 0-17 feet, attached to a hand- 
pump of D = 0-2 feet in diameter P= 44 pounds, and the end of the pipe ele- 
vated h = 20 feet above the piston Dt 



r =6-864 /^i 1 
V * 



(44 — 49X0-2^X20) __-,., 



1*95 feet per second. 



)-2(135+50X0-17) 

Q == 1-95X5-38X0-2* = 0-042 per secondX60 = 2-52 cubic feet per minute. 
* = 0-8 feet the stroke of piston, we shall have 
2-52 



r785*0-2* ~ 10 ° strokes P er m i nu te. 



" 0-8X0 



Table for Water flowing over Weirs* 

This Table is set up from careful experiments 
on a large scale, and is suited for weirs only. 
See Fig. 290. 

RULE. Multiply the width b in feet, of the 
weir by the coefficient k, and the product is the 
quantity of water discharged per second, in cubic 
feet, h is the height as represented by Fig. 290. 
The width b should be b > h. 

Example 6. How much water will flow over a 
wier of b = 5 feet, h = 0*5 feet in one minute ? 
Q = k b t = 1-1295X&X60 = 338-35 cubic feet. 



ft. inches. 


h.feet. 


m. 


k. 


0-4 


0-033 


0-424 


0-01365 


0-8 


0-066 


0-417 


0-05452 


1-2 


0-100 


0-412 


0-10592 


1-6 


0133 


0-407 


0-16616 


2-4 


0-200 


0-401 


0-29171 


3-2 


0-266 


0-397 


0-44480 


4- 


0-333 


0-395 


0-63111 


6- 


0-500 


0-393 


1-1295 


8- 


0-666 


0-390 


1-7464 


9* 


0-750 


0-385,2-0331 


12 


1-000 


0-376 T 


31350 



HYDRODYNAMICS. 

Water Power* 

The natural effect concentrated in a fall of water, is equal to the weight of the 
quantity of water passed through per second multiplied by the vertical space it 
falls. 

Fig. 297. Let Q be the quantity of water which passes through the orifice a in 
»the time t = 1" second, in cubic feet of 625 pounds each. 

h = the vertical space the water falls; then the value or natural effect of the 
fall is at the orifice a. 

P— 62-5 £ h, effects. 
But, ♦ Q = 5'06a\/h, then we have 

P^ZUbahyh. 
This will be in horse-power, 

'h* 



ff=0-573aft>^ 



H=0'llUQh t 



h = 



0-11340 



Example 1. In a creek passes 18 cubic feet of water per second. How high 
must that creek be dammed up to produce an effect of 10 horses ? 



fc = 



10 



0-1134X18 



= 4-9 feet, the answer. 



Water-Wheels. 223 



WATER-WHEELS. 

Water-wheels are of two essential kinds, namely, Vertical and Horizontal. 

The Vertical are subdivided into 

Over shot-wheels, Undershot-wheels, Bread-wheels, and High-breast and Low-breast 
wheels. 

The Hoi-izontal are with Floats, Screw-wheels, Turbin, Reaction-wheels, <£c. 

Waterwheels do not transmit in full the natural effect concentrated in a fall 
of water ; under most favourable circumstances 80 per cent, has been utilized, 
but under poor arrangements only 20 per cent, may be expected. 

Example 1. Fig. 302. The vertical section of the immersed floats of an under- 
shot-wheel in a mid-stream is a = 27 square feet, velocity of the stream V— 8*6, 
and v = 4 feet per second. Required the horse-power of the wheel H ~- ? 

JJ= ~(V— v)a = ^^(8-6 — 4)« = 11-4 horses. 

2lKT J 2l>0 v ' 

Example, 2. Fig. 307. On a breast-wheel is acting Q — 88 cubic feet of water 
per second, the head h = 8 feet, velocity of the wheel at the centre of the 
buckets v = 5 feet p_r second ; the water strikes the buckets at an angle u = 8° 
and velocity F = 7 feet per second. Required the horse-power of the wheel, 
H = l 



H== n^( 8+ ^ (7Xcos ' 8 °~ 5) ) = €5 



horses. 



Example 3. Required the effect of Poncelet's wheel, the head h = 4 feet, and 
the orifice a = 5 square feet, the velocity of the wheel at the centre of pressure 
of the floats is v = 6'78 feet per second ? 

F = 6-91 yl = 13-82 feet per second. 

Q = 6"5XoXy4 = 65 cubic feet per second. 

H= 6 °*^'' 8 (13-82 — 6-78) = 15'8 horses. 

Example 4. Fig. 309. A saw-mill wheel is to be built under a fall of h = 18 
feet, and to make n = 110 revolutions per minute. Required the proper diam- 
eter of the wheel. 

i> =tt? y 18"= 3-857 feet, 

at the centre of pressure of the buckets. 
Telocity F = 8)/18 = 33-94 feet per second. 

Velocity v = 3 ' 14 X3-857X110 = 22-2 feet per second. 

The fall discharged 30 cubic feet of water per second. Required the horse- 
power of the wheel. H = ? 

30V°2*2 
H= 200 r33 ' 9 " ~~ 22 ' 2 ) = S9 horses * 

How many square feet of dry Pine can it saw per hour? 

See page 150. 30X39 =1170square feet. 

The saw is meant to be applied direct on the wheel shaft 



224 



IffDRAULIC?.. 




302. 

Undershot wheel in a mid-stream. 

When V — 2v about, the effect will be, 

H= *J£, a = area of float. 
loOU 




303. 



Undershot- Wheel. 



When V = 2t>, about, # = -ff^' 




304. 



Poncelefs Wheel. 



H= ^(V — u), when A > 5 feet, 

Qv 

H = - Q _( V — v) when A < 5 feet, 

Q=*Smay/h, 7=6-91x^7 




Breast-Wheel with Parabolic drain. 



m 

Hydraulics. 



225 




( ( ( i K ^X 




C06. Low-breast Wheel. 

s - -n^~[ A + "35-( y cos*- v ) j 



Q = kb. V = -i. • See table for weirs. 
a 



307. Breast Wheel. 



2r -I&l> + -3&-< |re "»->] 





308. Overshot Wheel. 



Proper velocity about n = 35I) + 10Q > 
revolutions per minute. 



309. Saw-Mill Wheel. 

200 l ; 



Proper diameter of the Wheel, 

D=s ^.VX; in feet, 
n 

n = revolutions per min. 



226 



Turbine Wheels. 



TURBINES. 

Letters denote. 
Q = cubic feet of water passed through the turbine per second. 
h = height of fall in feet. 

D = diameter in inches of circle of effort in the turbine. 
a= area in sq. in. of the conduit passage into the turbine wheel. 
6 = depth in inches of turbine buckets. 
c = depth in inches of leading buckets. 
r = breadth of turbine buckets in inches. 
m = number of buckets in the turbine wheel. 
m'= number of leading buckets. 
n = number ot revolutions of turbine per minute. 
S and s=height of conduit and discharge in inches. 
t = thickness of steel plate buckets in 16ths of an inch. 
H = actual horse power of the turbine. 
I = length in feet ) _ f „-,„/!„.+ ^-„ rt 

d = diameter in inches } of conduit pipe. 
d' = diameter in inches of the discharge pipe. 

W= Hydraulic pressure on the turbine wheel bearing on the end of the 
shaft. 



D = 



fcyh 



D=- 



0*436 r 



- 2 



, kyh 

_20fe Q 

'~ aD ' 

a 
~~ 0-436D' 

_46fcQ 

D . D 

• = _ to—, 

5 8' 



- - - 3 



- 7 



: io' 




fl = 0-436 Dr, 
a = m f rs, - - 
a'=t mrs,- • 
a'=0-98 a, - 



Q 



_a Dn 

~ 20 k ' 



■ - 9 

- 10 

- 11 

- 12 

- 13 

- 14 

- 15 



- - 16 



m = 5 \Tj) i - 



0-625 D 
X m ' 

0-78 D 

\s s - - 
d'=D-|-2r, - - 



b = 



S = (H 



17 



19 



20 



22 
23 



pr=- 



H= 0-1134 Q y/fe natural effect of the fall, 

30 Q 3 

actual horse power, 

66 per cent of the natural. 



H = 



ahyH 



- 24 



25 



26 

27 



The coefficient k can vary from 800 to 1200 without seriously affecting 
the per centage of the ultilized power, but it is best between 900 and 1000. 
This is a great advantage of the turbine over water wheels, that under 
the same head of fall it can run at different velocities and still utilizing 
the maximum effect. Whatever coefficient ft adopted it must be kept the 
same throughout the construction of the turbine. 



Turbine Wheels. 



227 



Jonval's Turbine has so many advantages above other hydraulic mo- 
tors that it is considered sufficient to describe the construction of that 
one only, but the princippvl formulas will answer for any kind of turbines. 

On the accompanying plate is a drawing of a Jonval Turbine such as 
the Author of this Pocket Book has built in Russia. The buckets are not 
supported by concentric rings, but are fastened only on one side, which 
is considered more simple and convenient for replacing new buckets. For 
falls over 30 feet it may be better to make it with concentric rings. 
When a turbine is to be constructed we have on the one side given the 
natural effect of the fall, and on the other side the actual work to be 
done, which latter should not exceed 66 per cent, of the former. Between 
these two points the turbine is to be so proportioned as to utilize the 
greatest possible effect with smallest expense of Machinery. 

Jonval's turbine in good condition generally utilizes 60 to 80 per cent. 
Suppose a fall of ft =--25 feet, discharging Q=12 cubic feet of water per 
second, the natural etfect will be, 

H=0-1134X12X /25 = 6-8 horses, 
of which 6-8X0 , 66=4*5 horses to be counted upon as the actual effect of 
the turbine. 

Turbine shaft to make n=200 revolutions per minute with the assumed 
coefficient k =960. From these dates we will obtain all the principal 
dimensions of the turbine, namely, 



n 960 y 25 _. . , 

D= - — - = 24 inches, 

200 



20X960X12 



= 48 sq. in. 10 



24X200 

m = 5/24=24-5 say 25. - - 17 
m' = 4-5/24 = 22 buckets. - 18 



6 = 



0-436X24 

0*625X24 

j/25 



- = 4-6 in. 



= 3 in. 



- - 19 



^0^78X24 =4incheg> . . 2Q 

^22 

OK 

t = _ = 2-5, 16ths. - - - - 8 

10 ' 



In calculating the breadth r from formula 5, it must come inside of 
formula 7, if not the diameter D must be altered. 

Now proceed with the construction as shown at the bottom of the plate, 
which represents a section of the buckets through the circle of effort of 
the turbine. 

The drawing of the turbine is £ of an inch to the foot, and the construc- 
tion of the buckets 3 inches to the foot. 

Draw the base line AB, set off the angle of the leading buckets=10°. 
The distance between the leading buckets will in this case be 24X3-14:22= 
3-43 inches, set off this from S towards A, draw the straight part of the 
second bucket parallel to the first one, draw from S the line d d at right 
angle to the buckets, and e will be the centre for the curved part. From 
the centre of S draw the line o to the end of the second buckets, divide 
this line into eight equal parts take five of them as radus and draw from 
the end of the second bucket a circlearc of about 50°, which will be the 
propelling part of the turbine wheel bucket. 

Distance between the wheel buckets will be 24X3*14:25=3-02 inches, set 
off this from A towards S, draw the second propelling arc. Set off from A 
the depth of the wheel buckets 5=3 inches, set off 2 & to s, which will be 
the length of the first wheel bucket. Set off from s to u the distance 
between the buckets 3-02 inches. Make s=0-86 S. Draw from u a curved 
line in the form of a parabola that will leave the space s and tangent the 
propelling circlearc somewhere about x. Care must be taken that the 
discharging area a' of all the wheel buckets will be about 2 per cent, less 
than the conduit area a of all the leading buckets. The surface of the 
buckets should be made as smooth as possible, or even polished. 

For very high falls the Hydraulic pressure W becomes very considerable 



228 Water Pipes. 



and may necessitate another arrangement, namely, to lay the shaft hori- 
zontally and place on it two turbines so that the leading buckets are either 
betweeib or outside of the wheels, but then comes another disadvantage, 
namely, that the number of revolutions will be greatly increased and may 
be required to gear it down 10 to 20 times to the proper speed of the main 
shaft. 2) 

To avoid this as much as possible take fc=800 and make r=— . 

One great advantage with JonvaPs turbine is that it can be placed 
almost anywhere between the high and low levels to suit the location, 
though it should not be more than 20 feet above the lower level ; then in 
order to utilize the whole fall, care must be taken to make the discharge 
pipe perfectly air tight. It is not necessary to make the discharge straight 
down from the turbine, it can be carried horizontally or inclined, as may 
suit the location. The Author has built turbines similar to that repre- 
sented on the accompanying plate, at General 3Ialtzof s Establishment, 
Kaluga, Russia. 

Velocity of Water in Rivers, 

The velocity of the water at the bottom in rivers is to that at the sur- 
face, as 8 is to 10. 

MOTION OP WATER IN PIPES. 

For City Water works. Du BuaVs formula. 
Letters denote. 
0.= cubic feet of water passed through the pipe per minute. 
D = inside diameter of the pipe in feet. 
L = length of the pipe in feet increased by 50 diameters. 
H= differential head in feet. 
v = velocity of the water in the pipe in feet per minute. 

_ 2356y / D s £ = __L_ 6 / Q'L v = 3000_/5 

Q ~~Tl 22-329 \y H ' IL 

4fl N H 

Example 1. A water pipe of D=l*75 feet in diameter, L=36,000-|-50X1 , 75 
=36087-5 feet long, head pressure H=390 feet. Required how much water 
it can discharge per minute ] 



= 992-26 cubic feet. 



f- 



[36037-5 
390 " 

Example 2. At a distance of 27960 feet from a water work is required 
Q=564 cubic feet of water per minute, head pressure being H=25e'feet. 
Required the diameter of the pipe 1 L=27960-f-50-230l0 feet. 



1 5 / 
2-329 \/ 



22-329 \/ 256 

Example 3. A water pipe of D=0-75 feet in diameter, L=8650-f-50-=8700 
feet, have a head pressure of 11=128 feet. Required the velocity »=1 
of the discharge. 

5 =41-424 feet per second. 



j 



8700 
128~ 



Consumption of water in cubic feet per head of population, including 
all uses, as for manufactories, fires, &c, &c, in 24 hours. 
January, 2-58, I April, 2'73 | July, 4-58 I October, 4-46 

February, 2-40, May, 3-37 August, 4-75 November, 4*12 

March, 2.64, | June, 3*50 | Sept., 4-61 | December, 3*61 



JOHVAL'S TURBINE, 

as constructed by JohnW. Nystrom. 




230 



Atmosphere. Aerostatic. 



AtfMOSPHERE. AEROSTATIC. 

The atmosphere round our earth, as well as all other gaseous matters, en- 
deavours to occupy a larger space to infinity, (no known limits,) but as it is a 
material substance, it is under the action of force of gravity, and cannot expand 
farther than when its density is in equilibrium with the said force. Conceive 
the atmosphere to consist of a great number of layers, one on the top of the 
other; the density of the under layers will evidently be greatest, because the 
upper ones press on them, and they are all elastic ; hence the density of the 
atmosphere is greater at the surface of the earth than higher up. We can now 
find out the weight and density of til these layers. 



310. 




a! 



A is a vessel full of mercury, in which is placed verti- 
cally a glass tube about 3 feet high above the surface I; 
in the glass tube is fitted, air-tight, piston a, just one 
square inch area, which can be moved by the piston-rod 
c ; now the piston stand is at a on the level I, and in con- 
tact with the mercury in the tube ; raise the piston by 
the piston rod and handle c, the mercury in the tube will 
follow until the height of 30 inches, the piston still con- 
tinues to move higher in the tube, but the mercury will 
maintain its position at 30 inches from I. Now it may be 
supposed that it is some force of the piston that draws the 
mercury up in the tube ; if so why did it separate at 30 
inches ? If the column beeomes too heavy it could sepa- 
rate at Z, and the 30 inches of mercury follow the piston ; 
as this is not the case, but the weight of the atmosphere 
pressing on the surface I and forcing the mercury up in 
the tube until it (the mercury and the atmosphere) comes 
in equilibrium, which occurs at the 30 inches ; and the 
piston only served to remove the atmospheric pressure 
in the tube ; hence we have the weight of a column of 
atmospheric air with one square inch base equal to the 
weight of a column of mercury 30 inches high and one 
sq. in. base. One cubic inch of mercury at 60° Fahr. weighs 0'491 pounds, this 
multiplied by the height, 30 inches, gives 14*73 pounds, the weight of the col- 
umns of mercury or atmosphere ; this is generally termed " the atmospheric 
pressure per square inch." 
The specific gravity of mercury at 60° Fahr. isl3*58, and 



13-58X30 
12 



: 33*95 feet, the height of a column of 



water required to balance the atmosphere. 

If the temperature and force of gravity were uniform throughout the atmos- 
phere, the density would decrease in an arithmetical progression by the height 
from the ground, and by observing the altitude of columns of mercury at two 
different heights, the extreme height of the atmosphere would be found simply 
by the formula 7, page 64, in which 

a = o the altitude of the column of mercury at the top of the atmosphere. 
b = 30 inches, the column of mercury at the level of the sea. 
S = the difference of the columns of mercury at the level of the sea, and at a 
height h above the sea. 

Then, n multiplied by the height h, should be the extreme height of the 
atmosphere, or 



-(H 



Bahometer. 231 



Example. The mountain Chimborazo, Eucador, (South America,) is h = 3*87 
miles high above the level of the sea; at its top the column of mercury is ob- 
served to be only 27*63 inches, and S = 30 — 27*63 = 2-37. 



r = s - 87 (Jr7 +1 ) : 



■ 52-825 miles, the extreme height 

Of the atmosphere, 

This is about the true height, but the calculation is incomplete by lack of 
many circumstances accompanied with higher calcules in mathematics, which 
can not be allowed to occupy room in this work. 

The column is 30 inches when the temperature of the atmosphere is 32° Fah. 
(See Barometer.) 



311 . BAROMETER. 

The Barometer is based upon the same principle as the preceding experi- 
\ ment. It consists of a glass tube &, about 35 inches high, open at one 
„, end, c, and filled with distilled mercury ; inverted in a small vessel A, 
also containing mercury. About the top of the column of mercury is 
placed a scale to indicate the height from I. When disturbances take 
place in the atmosphere by heat, condensation, &c, its weight and den- 
sity will differ, and the column of mercury will fall and rise accordingly; 
hence by connection of the scale at a, it indicates the disturbance. 

To Find the Density of the Atmosphere about a 
Barometer^ 

5 LeMers denote. 

h = altitude of the column of mercury in inches. 
t = temperature of the atmosphere, Fah. 

S — specific gravity of the air around the Barometer at the time h and 
t are observed. 

£=1 when £ = 32°. 

* * 2 

* ~ 0-0624(448-77 +<) ** 

Example. The Barometer has fallen to 26*31 inches and the tem- 
perature is t = 60°. Required the specific gravity of the air ? 
26-31 



' 0-0624(448-77+60) 



= 0-83. 



This formula does not include the expansion of mercury from 32° to 60°, 
which must be separately reduced by the following formula. 

71 = ^ (0-9967962 — 0-00010010 .... 3, 
in which 2T= the observed column at the temperature t, and h = the true 
column to be inserted in the formula 2. 

To Measure Vertical Heights "hy the Barometer* 

Letters Denote, 

I = latitude of the place. 
f = vertical height, in feet, between the higher and lower station. 



232 Wind. Areodtnamio. 



x ^AA^i+o-oooiooitr— t))' - - - - 4, 

/=60345-51a;(l+0-002551cos.2Z)(l+0'00208(r+« — 64°)), - - 5. 

If the atmosphere is very calm the observations may he made one after the 
other by one Barometer and detached Thermometer ; but the least disturbance 
of wind requires the observations at the upper and lower stations to be made 
at the same time. The reduction of the columns of mercury is included in the 
formula 5. 



WIND. AREODYNAMIC. 

The motions and effects of gases by the force of gravity, are precisely the same 
as that of liquids. (See Hydraulics.) 

The altitude or head of the atmosphere at uniform density will be the alti- 
tude of a column of water 33-95 feet, divided by the specific gravity of the air, 
0-0012046, or, 

1 . = 28183 feet, 



0-0012046 
the velocity due at the foot of this head is (Formula 1, page 183.) 

F= 8-02 >/2S183~= 1346*4 feet per second, the velocity at which the air will 
pass into a vacuum. 

Velocity of Wind. 
When air passes into an air of less density, the velocity of its passage is mea- 
sured by the difference of their density. 

"v ~ > density of the air in inches of mercury. 

t = temperature at the time of passage. 
V = velocity of the wind in feet per second. 



'= 1346-4 * / ^-^(l+0-00208A, 



6. 



The force of wind increases as the square ofiits velocity. 

a = area exposed at right-angles to the wind, in square feet. 

F= force of the wind in pounds. 

H= horse-power. 

v = velocity of the plane a in direction of the wind, + when it moves oppo- 
site, and — when it moves with the wind. 

F= 0-002288a F», when v = o, - • 7, 
.F=0-002288a(F:p)9, 8, 

241400 ' '9. 

Example. A Rail-train running ENE 25 miles per hour, exposes a surface of 
1000 square feet to a pleasant brisk gale NE by E. Required the resistance to 
the train in the direction it moves, and the horse-power lost ? 
E]VE—JVEbyjSr=S points = 33° 45'. 
V= 14 feet per second, a brisk gale. 
v = 25X1*467 = 36-6 feet per second. 
F= 0-002288 sin.*33° 46'Xl000(14+cos.33° 45'X36-6)* = 3051 pounds. 

„ 305-1X36-6 . , 
JT= ■ — —> — = 20 horses. 
550 



Table op Velocity and Force op Wind.— Balloon. 



233 



Miles 


feet per 


Force per 


r hour. 


second. 


sq.ft pound. 


1 


1-47 


0-005 


2 


2-93 


0-020 


3 


4-4 


0-044 


4 


5-87 


0-079 


5 


7-33 


0-123 


6 


8-8 


0-177 


7 


10-25 


0-241 


8 


11-75 


0-315 


9 


13-2 


0-400 


10 


14-67 


0-492 


12 


17-6 


0708 


14 ' 


20-5 


0-964 


15 


22-00 


1-107 


16 


23-45 


1-25 


18 


26-4 


1-55 


20 


29-34 


1-968 


25 


36-67 


3-075 


30 


44-01 


4-429 


35 


51-34 


6-027 


40 


58-68 


7-873 


45 


66-01 


9-963 


50 


73-35 


12-30 


55 


80-7 


14-9 


60 


88-02 


17-71 


65 


95-4 


20-85 


70 


102-5 


24-1 


75 


110 


27-7 


80 


117-36 


31-49 


.00 


146-66 


50- 



Common Appellation of the Force 

Wind. 
J Hardly perceptible. 
Just perceptible. 

Gentle pleasant wind. 



Pleasant brisk gale. 



-Very brisk. 

f High wind. 

Very high. 

Storm or tempest. 

Great storm. 
\ Hurricane. 
Tornado, tearing up trees, &c. 



BALLOON. 



To Find what Weights and to what Height a Balloon can 
raise* 

Letters denote. 

C= cubic contents of the balloon, in feet. 

& = specific gravity of the gas used to inflate the balloon, air = 1 at 32°. 
W— the weight in pounds, it can raise from the ground. 
w = the weight with which it is loaded, including the weight of the materials 
of which it is made. 
f = height in feet to which it will raise. 
T= temperature at the ground. 
t = temperature -et the height/. 
H = Barometer column in inches, at the ground. 
I = latitude of the place. 

TT== 0-07529(7(1 — s), 10, 



s = log./' 



W ^1+0-OOOlOOlCT 



=») 



11, 



12. 



/= 6O345-51z(l-f-0-002551 cos.2Z)(l-f 0'00208(F+* — 64) ) 
Balloons are commonly filled with Hydrogen gas whose specific gravity ia 
s = 0-07, when pure, or about 14 times lighter than air, of which say 10 times to 
he relied upon, as some foreign heavier gases may accompany it. 



20* 



234 Wind-Mills. 



WIND-MILLS. 



The sail-shaft of vertical wind-mills should have an inclination from 12° to 
15° with the level when "built on low flat ground ; on high ground, elevated 
from 1000 to 1500 feet within a circle of about two miles, the sail-shaft should 
incline from 3° to 6° with the level 

Effect of Wind-Mills. 
Letters denote. 
A = projecting area of sails exposed to the wind, in square feet. 
V = velocity of the wind in feet per second. 
H= horse-power of the iuilL 

^ = fnuer me } radiiofsai:siafeet - 

I = V - " rr -, radius of centre of percussion in feet. 

91 = number of revolutions of sails per minute. 
v = mean angle of sails to the plane of motion. 
The angle of the sails should be from 20° to 30° at the inner radius r, at the 
extreme radius M from 7° to 12°, and the mean angle v = 15° to 17°. 



H= 



A I n sin.v cos.v / „ b n sin.v 



/ y b n sin.v \* 



1,540,000 
assume the mean angle v = 16°, we have the horse power. 



H= 



A 



OV 34-5 ' 



5,800,000' 

In order to utilise the maximium effect of wind, it is necessary to load the 
mill that the number of revolutions of the sails are proportional to the velocity 
of the wind. 

Proper revolutions will be found by n =— — . . 

I sin. v 

T*« mo „ 11-5 y tt Ay% ot^ A 1,135.000 g 
Ift,=16o, TC = T _, B-—— 9 and 4—^ 

Example 1. A wind-mill is to be built of six horse power in brisk wind, 
Fj= 20 feet per second. Required the area of sails A = ? 
^1,135.000X6 =851 fee4 , 
203 
851 
Example 2. Four sails — - — 212-75 sq. feet each. 212-75 = 6 feet wide by 

35*5 long, dimensions of the sails. Inner radius r = 5 feet and R = 5+35*3 
=40*5 feet. Required the radius of centre of percussion 1 = 1 

l=s ^/ 40-5*4-5 3^ v /832 7 B~=28-85feet. 

Example 3. The mean angle of sails to be v = 16°. Required the proper 
number of revolutions of the sails per minute in brisk wind of V = 20 ieet yer 
second n, — ? 

Revolutions n =!l£^= 8 per minute. 

Example 4. A wind-mill has an area of A = 750 sq. feet exposed to high wind 
of V— 50 feet per second, and makes n = 26 revolutions per minute,— centre 
of percussion I = 25. Required the horse power of the mill H= t and proper 
number of revolutions per minute n = ? 

„_ 750X25X26 / 25X27 y = M norges# 
V 34-5 / 



5,800,000 V 34-5 

» = li^i°=23rev. P 

750X25X23 ( 5 q^J5X23 N* = g . 26 horses# 
5.800.000 V 34-5 / 



n== ll-5X50 = 23 rey> per minute. 
25 



Gas, Light jihd Sound. 235 



LIGHT. 

Iiiffht is the sensation transmitted by the eye and produces the sense 
of seeing. Heat and Electricity produce light by making bodies luminous. 

Intensity of Light is inverse as the square of the distance from the lumi- 
nous body. 

Velocity of Light is 192500 miles per second. 

Light passes from the sun 95000000 miles in 8 minutes. 

Light can pass around the whole earth in one-eighth of a second. 

Solids must be heated to at least 600° to produce light in the dark ; and 
o 1C00° in day-light. 

MOTIONOFGASINPIPES. 

Letters denote. 
Q = cubic feet of gas passed through the gas pipe per hour. 
L = length in feet, D=diameter in inches of the pipe. 
H= head of water in inches which presses the gas through the pipe. 
s — specific gravity of the gas, air being 1. 

n = number of candles required for giving the same light as Q cubic feet 
of gas per hour. 

Example. At a distance of L=6450 feet from the gas work is required 
Q=940 cubic feet of gas per hour. Head of water being H=Q inches, 
specific gravity s=0-5. Required the diameter of the pipe D= 1 



V SL' „ =q3 14^35 V « 

ance c 
?as p< 
i. Re( 



D== _^_ v 7 Q-aXMSOXW = 3 . 7837 lnches , 

14-35 \/ 6 



SOUND. 

Velocity of Sound through Air, 

v = velocity in feet per second. 
t = temperature of the air, Fah. scale. 
D = distance in feet the sound travels in the time T. 



v = 1089-42 ya-t-0-0020S^ — 32), 
Telocity of sound in water is about 4 times that in air, and 8 times that 
through solids. 
Intensity of sound is inversely as the square of the distance. 



D = 1089-42T >/l+0 00208(i — 32), 

r-£. 

V 

Example. A ship at sea was seen to fire a cannon, and 6*5 seconds afterwards 
the report was heard, the temperature in the air was 60°. Required the dis- 
tance to the ship ? 

D = 10S9-42X6-5 j/l+0-002S(60° — 32) = 7300 feet, or 1-38 miles. 



Descriptions of Sound, 

A powerful human voice in the open air, no wind, 

Report of a musket, 

Drum, --------- 

Music, strong brass band, 

Cannonading, very strong, 

In a barely observable breeze a strong human voice 
with the wind can be heard, - - 



Audible at a distance of 



feet. 


miles. 


460 


C-087 


16000 


3-02 


10500 


2 


15S40 


3 


575000 


90 



15S40 



236 Bells. 



RINGING BELLS. 



Letters denote. 

D = diameter of the bell at the mouth, in inches. 

d = diameter of the bell at the crown, in inches. 

h = heighth of the bell from the mouth to the crown in inches. 

S = thickness of sound bow iu inches. 
W= weight of the bell in pounds avoirdupois. 

n = number of vibrations per second, corresponding with the key note of the 
boll, and to be found in the accompanying table I. 

k = from 0-07 to 0*08, or a coefficient expressing the relative thickness of the 
sound bow to the diameter of the bell. In peals of bells, the sound bow is 
generally S -= 0082) for the triple, and S = 0-072? for the tenor ; the interme- 
diate bells in the peal having the intermediate proportions of sound bow. 

Example 1. Required the weight of a bell D = 62 inches in diameter, and £ 
= 4$in. thickness of the sound bow, W= ? 

Formulae 1. W= 0-25X622X4-5 = 4324-5 pounds. 

Example 2. A bell of 2,500 pounds is to be constructed with a sharp note, 
taking the sound bow k = 0*075. Required the diameter of the bell D — ? 



Formulae 10. D = V t^^ = 51-084 inches 
00<5 

Example 3. It is required to construct a bell with the key note Z>j} in the 
first octave above zero, n = 152-22. To be of light weight with a full good note, 
for which latter case take k = 0*07. Required the diameter of the bell, D = ? 

Formulae 11. D = 5 ??? * ' 07 - 26-665 inches. 

Example 4. Required the key note of a bell with D = 36*5 in. diameter and 
S = 2-75 in., n = ? 

2*75 
Formulae 4. n = 58000X 5*-?- = 119 ' 7 vibrations. 

OU.D 

Tn the table the nearest number 120-82, in the first octave below zero, answers 
to the key note 2?, which will be the note of the bell. 

Example 5. A bell of 6860 pounds is to be constructed with the key note C 
in the first octave below nero n = 64, see table I. Required the diameter of 

the bell D = ? 

4 /6860 
'Formulae 9. D — 21-947 L / -gj- = 70*6175 inches 



47 * r- 

V' 



Example 6. Required the thickness of sound bow for the bell in the pre- 
ceding example ? D = 70.6175 inches and n = 64. S = ? 

Formulae 12. S = ~ 4X J°^ 75 * = 5-5027 inches. 

OoOUO 

Example 7. Required the weight of a bell D = 48 inches diameter at the 
mouth, d = 25 inches at the crown, and h = 34 inches height from the mouth 
to the crown, S= 35 in., W= ? 

Formulae 17. 
W*= 48X25X3*5 (0-5—0-002816X25) +0*00375X34X252X3-5 = 2126-226 pounds. 



BELL3. 



Formulas for Ringing Bellso 



237 



W= 0-25 D*S - 

232000 
W = 0-25Z8& - . 



n = 58000- 
n = 232000 






J?^ 



if 



* /If 
3 /Tw 

2> =58000-^ - - - 



D = 240-83 
4 D = 21-947 

5 



7 S = 
8£ = 



58000 " 
4TF 









= pa 



TF= D d S (0-5—0-0002816 d) +0-00375 h d* S 



Ta&Ze 7. Vibrations per Second — n. 



Key 




Bass. 


< 


9 


Descant. 




note. 


3rd Oct. 


2nd Oct. 


1st Oct. £ 1st Oct. 


2nd Oct. 




C 


16-000 


32-000 


64-000 


128-00 


256-00 


512-00 


£* 


16-947 


33-385 


67-790 


135-58 


271-00 


542-32 


17-960 


35-920 


a-840 


143-68 


287-36 


574-72 


? 


19-027 


38-055 


76-110 


152-22 


304-44 


608-88 


20-159 


40-318 


80-636 


161-27 


322-54 


645-09 


P 


21-357 


42-715 


85-430 


170-86 


341-72 


683-44 




22-627 


45-255 


90-510 


181-02 


362-04 


724-08 


23-972 


47-945 


95-890 


191-78 


383-56 


767-12 


A* 


25-398 


50-797 


101-59 


203-19 


406-37 


812-75 


26-908 


53-817 


107-63 


215-27 


430-53 


861-07 


a8 


28-508 


57-017 


114-03 


228-07 


456-13 


912-27 


B 


30-204 


60-409 


120-82 


211-63 


483-27 


966-54 


C 


32-000 


64-000 


128-00 


256-00 


512-00 


1024-0 



Talle V. 



Abscissa 


Ordinate 




Thickness of Metal. 




X 


y 


£=1 


£=0-07Z> 


S = 0-75Z> 


£=0'08I> 


1 


0-4142 


1 


0-700 


0-750 


0-800 


H 


0-686 


0-800 


0-560 


0-600 


0.640 


2 


0-867 


0-653 


0-459 


0-490 


0-522 


2± 


0-974 


0-547 


0-382 


0-410 


0-437 


3 


1-025 


0-474 


0-331 


0-355 


0-379 


3* 


1-030 


0-423 


0-295 


0-317 


0-338 


4 


1-000 


0-380 


0-266 


0-285 


0304 


4* 


0-955 


0-351 


0-245 


0-263 


0-281 


5 


0-875 


0-327 


0.228 


0-245 


0-261 


5*. 


0-775 


0-301 


0211 


0-226 


241 


6 


0-665 


0-291 


0-203 


0-218 


0-233 


6| 


0-530 


0-286 


0-200 


0-214 


0-228 


7 


0-390 


0.279 


0-195 


0-209 


0-223 


71 


0-235 


0.272 


0-190 


0-204 


0-217 


8 


0075 


0.267 


0-186 


0-200' 


0-213 


8-74 


0-78 


0.333 


0.233 


0.250 


0-266 



238 



Bells. 



To Construct a Bell. 

When a bell is to be constructed, we generally have the weight or key- 
note given by contract, the diameter and sound bow are calculated by 
the preceding formulas and examples, and then ready to proceed with 
the construction. See fig. 1. 

The diameter of the bell at the mouth, is divided into 10 equal parts, 
called strokes, which then is the scale' and measurement for the con- 
struction. Make a decimal scale, as shown on plate VII. 

Shrinkage to be allowed for 3 sixteenths of an inch per foot. 

The section of a bell is generally laid out on a piece of board repre- 
sented by the dotted lines a, b, c, d, which then is cut out and used for 
turning up the mould for the bell. The board should be about 11 strokes 
long, and 2-5 strokes wide. Through the centre of the board draw the 
lineup, q, parallel to &, c, bisect the line 2?, q, and set four (4) strokes from 
the bisecting point towards each end, divide the strokes into halves, and 
number them as shown on the accompanying drawing. Through each 
division draw lines at right angles to p, q, set off the corresponding ordi- 
nates y expressed in strokes, Table II. and join them by a curve-line, 
which then will be the centre of thickness of metal in the bell. 

At the end of the first ordinate, as a centre, draw a circle with a diameter 
equal to the desired thickness of the sound bow, which should be from 
0*7 to 0*8 strokes. At every succeeding ordinate draw a circle with the 
diameter noted in Table II ; for instance, if the thickness of the sound 
bow is 4£ inches, then the thickness of metal or diameter of the circle at 
the third ordinate will be 4-5X0-474=2-133 inches ; but if the sound bow 
is 0-7, 0-75 or 0*8 strokes, the thickness of metal at the third ordinate will be 
0*331, 0-355, or 0-379 strokes. When all the thicknesses are thus drawn, 
draw the two lines tangenting the circles on each side of the centre line 
of the metal. 

From to 1 make a moulding of 0-1 stroke thick over the dotted line 
as shown by fig. 2. 

Prolong the 6k ordinate, and set off 1-79 strokes to e, which then is the 
centre for the curve on the top, draw the arc through the centre of the 
small circle at the 8th ordinate ; join c, 8, set off from e, 0-46 strokes to 
the centre for the inside curve at the top. 

Thickness of metal of the top should be 0-3 the sound bow at 8, and 
0*333 at r. Draw the ordinate at 8-74, set off 0-78 to r, join r and the abscissa 
8-48, and prolong the line through r; then finish the drawing as shown on 
the plate. 

When the board is cut out and ready for turning the mould, it must be 
carefully set, so that the outside diameter of the crown will be half the 
diameter of the mouth of the bell. 

This form of Bells gives the greatest possible gravity of tone with the 
least possible quantity of metal. Bells can be made almost in any form 
without seriously affecting the quality of tone, but the thickness of metal 
should always be in proportion as the square of the diameter taken at the 
centre of the metal as in fig. 3. 

Proportions of a Peal of Eight Bells, 

Bells. 



Keynote. 


n 


k 


S. in. 


1 


D 


71-84 


0-070 


3-95 




E.. 


80-64 


0-071 


3-62 




4 
G ' 


90-51 


0-072 


3-32 




95-89 


0-073 


3-22 




A 


107.63 


0-075 


3-08 




B 


120-82 


0-077 


2-85 




4 


135-58 


0-079 


2-67 




143-68 


0-080 


2-58 





Tenor, 

2nd, 

3rd, 

4th, 

5th, 

6th, 

7th, 
Triple, 

Clapper. The weight of the clapper should be from one fortieth to 
one fiftieth the weight of the Bell, the smaller bells takes the largest 
clappers. 

Bell Metal. Thirty of Tin to one hundred of Copper, is a good pro- 
portion. 



D. in. 


W. lbs. 


Clapper. 


56-5 


3156 


63 lbs. 


51-1 


2366 


4S-6 


46-1 


1765 


37-2 


44-2 


1575 


34-1 


40-5 


1262 


28-1 


37-0 


976 


22-4 


33-8 


763 


18-2 


32*3 


673 


16-8 



] 'lA •Ir.YII 



s mMMW& BISIS, 



Fio.l. 




Music. 




240 



Heat. Caloric. 



HEAT. CALOEIC. 

The Physical constitution of heat is yet under investigation by operative minds, 
its well known character and effect upon matter is the base for the investiga- 
tion. 

Heat resembles light, electricity and magnetism, and is thus far assumed to be a 
material substance. 

Heat is contained in all matters, with no known exception. Two bodies con- 
taining different quantities of heat per unit, being placed in contact, — the heat 
will pass from one to the other until it comes in equilibrium, that is when 
the two bodies contain equal quantities of heat per unit. It is this passing of 
heat that first comes under our notice. The body from which the heat passes 
will feel the other to be cold, and vice versa. — the one that receives the heat will 
feel the other to be warm, until there is no further passage, namely, when the 
bodies will feel neither warm nor cold to each other ; hence the measure of the 
emplix sensible heat, is the difference between the heat per unit in the two 
bodies. Cold is only a want of heat. 

Caloric is only another word of expression for heat. 

Caloric is of two kinds, sensible and latent. 

Sensible Caloric is that which is sensible to the touch, felt as tempera- 
ture, and can pass freely from one body to another. 

Latent Caloric is that which is insensible to the touch. It is contained 
in bodies without being felt as temperature, but can by chemical action become 
sensible ; for instance, a piece of burned limestone put into water will get warm 
and heat it, although both were cold before ; the latent caloric in the water was 
set free to sensible. 

Influence of Heat on Matter's Coherence- 
All bodies in nature expand when heated, and contract when cooled. Solid 
bodies vary but little by the difference in temperature. Liquids vary more, but 
gases are extremely susceptible to the impression of heat and cold. 

Table of Linear Expansion of Solids. 



Difference in 
temperatures, 
32° to 212° 
32 to 392 
to 572 
to 212 
to 572 
to 212 
to 212 
to 212 
to 212 
to 212 
to 212 
to 572 
to 212 
to 212 
to 212 
to 212 
to 212 
to 212 
to 212 
to 212 
to 572 
to 572 
to 212 
32 to 212 
32 to 212 
32 to 212 

32 to 212 

33 to 212 
32 to 212 



32 
32 

82 
32 
32 
82 
32 
32 
32 
32 
32 
32 
82 
32 
82 
32 
32 
32 
32 
32 
82 



Length = 1 at 32°. 
Names of bodies. 

> Glass, ^ - 

[■ Wrought iron, 

Soft good iron, 

Iron wire, 

Cast iron, 

Soft steel, 

Steel hardened and tern. 150°, 

\ Copper, 

Lead, - - 
Gold, pure, - 
Gold, hammered, - 
Silver, pure, - 
Silver, hammered, 
Brass, common cast, 
Brass wire or sheet, 

< Platinum, pure, - 

Platinum, hammered, 
Zinc, pure or cast, 
Zinc, hammered, - 
Tin, hammered, - 
Tin, cast, 
Fire brick, 
Marble, - 
Granite, - 



Length at T.° 

1-00086133 
1-00184520 
1-00303252 
1-00118210 
1-00440528 
1-00122045 
1-00123504 
1-00111120 
1-00107915 
1-00123956 
1-00171820 
1-00564972 
1-00284836 
1-00146606 
1-00149530 
1-00190868 
1-00201000 
1-00187821 
1-00193333 
1-00088420 
1-00275482 
1-00095420 
1-00294107 
1-00310833 
1-0027' WOO 
1-00217298 
1-00042280 
1-0011 G410 
1-00078940 



k difference length 
per degree. 
0-00000478 
0-00000546 
0-00000583 
0-00000656 
0-00000894 
0-00000680 
0-00000687 
0-00000618 
0-00000600 
0-00000689 
0-00000955 
0-00001092 
0-00001580 
0-00000815 
0-00000830 
0-00001060 
0-00001116 
0-00001043 
0-00001075 
0-00000491 
0-00000520 
0-00000530 
0-00001633 
0-00001722 
9-00001500 
0-00001207 
)-00000235 
5-00000613 
00000438 



Heat. Caloric. 



Table of Volume Expansion of Liquids* 



Difference in 
temperatures. 


Karnes of Liquids. 


Yohcme at TP 


k difference in vol. 
per degree. 


32° to 212° 


Mercury, 


1-018018 


o-uooiooo 


212 to 392 


« * • m 


1'018133 


0*0001025 


392 to 572 


U m m 


1-018868 


'0001048 


32 to 212 


Water, - 


1-016600 


0*0002595 


32 to 212 


Salt, dissolved, - 


1 '€50000 


0*0002778 


32 to 212 


Sulphuric acid, - 


1*060000 


0*0003333 


32 to 212 


Oil of Turpentine and Ether, 


1-070000 


0* 0003S90 


32 to 212 


Oil, common, 


1-080000 


0*0001444 


32 to 212 


Alcohol and Nitric acid, 


1-100000 


0*0005555 



All gases expand and contract equally and uniformly ; 0*0020825 its volume per 
degree of Fah. thermometer. The accompanying Table is the result of Mr, 
Dalton's experiments with air. The volume at 32° is equal to 1 or the unit. 

Table for Volume Expansion of Air. 



Degrees. 
32° 
33 
34 
35 
40 
45 
50 
55 
60 
65 
70 
75 



Volume. 
1-000 
1-002 
1-004 
1-007 
1-021 
1-032 
1-043 
1-055 
1-066 
1-077 
1-089 
1-099 



Degrees. 

80 

85 

90 
100 
200 
212 
302 
392 
482 
572 



Yolume. 

1-1110 

1121 

1132 

1-152 

1-354 

1-376 

1-558 

1-739 

1-919 

2-098 

2-312 



Volume 

l 

t 



i? 



1 ] 
ItV 

1 -2S 

2 A 

2tV 



Letters Denote, 
L = length or any linear measure of the body of Mie temperature T. 
I == length or linear measure at the temperature t. 
V= volume of liquids at the temperature T: 
v = volume at the temperature t. 

h = coefficient for the linear measure or volume as noted in the Tables. 
The volume of solids is as L 9 : I*. 
The linear measure of liquids is a £'v : &Y\ 

Formulas of Linear Expansion of Solids* 

L = l(l+Jc(T—t)\ 



Z = 



*.-$?«• 



*= r— 



z— i 



ki> 



Example 1. A copper rod of L = 22-55 feet long is 140° warm. To what tem- 
perature must it be cooled to fit in a space of I = 22-52 feet ? 



t = 140 — 



22-55 — 22-52 
22-52X0-0000158 ' 



= 55-7° the answer. 



21 



242 



Heat. Thermometers. 



Formulas of Volume Expansion of Liquids* 



r =»A+fccr-A 



t=T— 



T- 



. kv 



Example 2. A vessel containing 5*68 cubic feet of water at t = 42°, is closed 
up round the water, but a cylindrical pipe of 0-008 square feet, inside section, is 
raised up vertically from it ; now let the temperature of the water be raised to 
T = 130°. How high will the water rise in the pipe ? 

F = 5-68 [1+0-000002595(130 — 42)] = 5*681297 cubic feet, 
5-681297 — 5-68 



and 



0-008 



- = 0162 feet = 1'945 inches, 



the height to which the water will raise in the pipe. 

This is the principle upon which Thermometers are constructed, but the scale 
can only be approximated by this formula. The substances adopted for ther- 
mometers are spirits of wine and mercury ; oil and ether has also been proposed, 
but the two former are best, and mercury is most generally used. 



THERMOMETERS. 

There are three different graduated Thermometers in use, namely Fahrer*- 
heifs Celcius's, and Reamur's. 

The first one, or Fahrenheit's is used in North America, England, and Holland. 

The second or Celcius's in France, Sweden and Germany. 

The third one, or Reamur's, was formerly used in France and some parts of 
Germany, but now only in Spain. 

The Figures exhibit their difference. 
Falir. Celci. Ream. 

Proportional Formulas for the 
Therniometrical Scales* 

Celci. = | Ream. = f (Fah. — 32.) 

Ream. = f Celci. = | (Fah. — 32.) 

Fahr. = | Celci. +32 = f Ream.+32. 

Example. How much is 6S° Celcius on Fahren- 
heit's scale. 

Fahrenheit's = |X68+32 = 154*4°, the answer. 



Fluid boils when its vapour has the same density as the atmosphere where it 
boils, hence, fluid will boil sooner high up in the atmosphere than at the 
ground. 

In vacuum water boils at 88°. 

The mean temperature of the earth is about 50°; at the torrid zone 75°' 
temperate zone 50° ; and in the polar regions 36°. 

Water can be kept in liquid to 20°. 










Taele of ' 


rEMPEKATUItES. 


243 


Table of Temperatures wb.cn Bodies cliange Form» 


SMELTING POINTS. 


BOILING POINTS. 




Cast iron, fully sm., - 


2754° 


Mercury, ... 


630° 


Gold, fine, • 




1983° 


Oil of Linseed, 


600° 


Silver, fine, - 




1850° 


Sweet Oil, ... 


412° 


Copper, . 




2160° 


Sulphuric acid, - 


410° 


Brass, common, - 




1900° 


Sulphur, ... 


390° 


Zinc ; ... 




740° 


Phosphorus, 


374° 


Lead, - 




594° 


Oil of turpentine, 


315° 


Bismuth, 




470° 


Sea-water, salt, - 


217° 


Tin, .... 




421° 


Water, distilled, • 


212° 


1 Tin, 1 Bismuth, 




283° 


Alcohol, ... 


174° 


3 Tin, 2 Lead, 5 Bismuth, 
1 Tin, 1 Lead, 4 Bismuth, 


212° 
201° 


MISCELLANEOUS. 




Antimony, - 


790° 


Metals, red, daylight, - 


- 1077° 


Sulphur, - 


228° 


Iron red, daylight, 


884° 


Phosphorus, - 


109° 


Common fire, 


790° 


Beeswax, white, - 


155° 


Iron bright red, in dark, « 


752° 


" yellow, - 


142° 


Human blood is - 


98° 


Tallow, - 


92° 


Cold greatest ever producec 


, —90° 


Ice, - 


32° 


Yenous fermentation, - — ■ 


- 60 to 70° 


Oil of Turpentine, 


14° 


Acetous fermentation begins, — 78° 


Ice of strong Brandy, 


7° 


Acetification ends, 


— 88° 


1 Snow and 1 Salt, 




Phosphorous burns, - 


— 43° 


Mercury, - 


- —39° 


A comfortable room about 


60° to 70° 


Table of Power for T 


ransmission of Heat* 






Sewing-silk, 


0-917 


Gold, - - - 1000 


Air, ... 


0-577 


Silver, • 


973 


RELATIVE CONDUCTING POWER OF 


Iron, ... 


347 


FLUIDS. 




Tin, .... 


304 


Mercury, ... 


1000 


Copper, ... 


898 


Water, 


357 


Zinc,- ... 


363 


Proof spirit, 


312 


Lead, ... 


180 


Alcohol pure, 


332 


Platinum, • 


981 


RADIATING POWER. 


Marble, ... 


24 


Water, 


100 


Fire-brick, - - - 


11 


Lampblack, ... 


100 


Fire-clay, ... 


11-4 


Paper, writing, 


98 


Porcelain, ... 


122 


Rosin, • 


96 


Water as the Standard, 


Sealing wax,' 


96 


Water, - 

Pine, 

Lime, - 


10 
39 

39 


Glass, common, 

India ink, ... 

Ice, - 

Red Lead, • 


90 

88 
85 
80 


Oak, - - - - 
Elm, 

Ash,- 

Apple, ... 


33 
32 
31 

28 


Graphit, ... 


75 


Lead, tempered, 


45 


Mercury, ... 
Lead, polished, 


20 
19 


Ebony, ... z^ 


Iron, polished, 


15 


RELATIVE CONDUCTING POWERS 


Tin and Silver, 


12 


OF SOLIDS. 


Copper and Gold, - 


32 


Hare's fur, - 




1-315 


REFLECTING POWERS. 


Eider-down, 




1-305 


Brass, ... 


100 


Beaver's fur, 




1-296 


Silver, 


90 


Raw silk, - 




1-284 


Tinfolium ... 


85 


Wool 




1-118 


Tin, .... 


80 


Lamp-black, 




1-117 


Steel, ... 


70 


Cotton, 




1-046 


Lead, ... 


60 


Lint, 




1-032 


Glass, ... 


10 


Charcoal, - 




0-936 


Glass, oiled or waxed, 


5 


Ashes of wood, - • 0-927 


Lampblack, ... 






244 



Specific Caloric. 



Mixtures of. 
Nitrate of Ammonia, 
Water, 

Sulphate of Soda, 
Muriatic Acid, 
Dilute Sulphuric Acid, 5 ) 
Snow, 4j 



Cold produce. 

1} i6 ° 

50° 



1} 



23° 



Degrees Fdhr. 
From-r-50° to -f<i°. 

From -{-50° to +0°. 

From — 68° to— 01°. 



SPECIFIC CALORIC. 

Specific Caloric is the relative quantity of heat contained in bodies of equal 
weight or volume, and of the same temperature. 

Let two different substances of known weight or volume and temperature, be 
mixed together; the temperature of the mixture will dissolve the relative 
quantity of caloric in the ingredients. 

Mixture of tne same Substances* 

betters denote. 
JV= weight or volume of a substance of temperature T. 
w = weight or volume of a similar substance but temperature U 
V = temperature of the mixture W-\-w. We shall have, 

WT+wt 



t'(W+w) = WTj-wt, 



W-- 



w(t' — t ) 
'' T—t fi 



W-\-w 9 
T _w(t'-t) 



Example 1. Let W= 4*62 cubic feet of water at T = 150° be mixed with 
w = 5*43 cubic feet at t = 46. Required the temperature of the mixture i' = ? 

,, 4-62Xl50°+5-43X46 o „ aQ .. 

f = 4-62+5-43 " 97 ' 6 the anSWer * 

Example 2. How much water of T = 107° must be mixed to w — 27*3 gallons 
of i — 58°, the mixture of the water to be 75° ? 



W-- 



27-3(75 — 58) 
107 — 75 



= 14-5 gallons. 



Mixture of different Substances* 

TPand w expressed by weights only. S and s = Specific caloric as given in 



the accompanying Table. We shall have, 
WS{T— t') = ws(t' — f), 



WS T+io s t 
WS+ws ' 



W= 



w s(t'— t) 
S(T-t)> 



_ t'{ WS+i v s) --wjj 
1 WS 



Example 3. To what temperature must TT= 20 pounds of iron be heated to 
raise w = 131 pounds of water of t = 54° to a temperature V = 64°. T = ? 
From the Table we have s = 1. and S = 0-1218. 



T = 



64(20X0-1218+131)— 131X1X54 
20X0 7 1218 



= 602°. 



the required temperature, supposing no vapour escapes from the water. 

If any chemical action takes place in the mixture, these formulas will not 
answer, because part of the sensible caloric may become latent, or latent caloric 
may be set free. 



Table of Specific Caloric. 



245 



Table of Specific Caloric, Water as Unit. 



Names of Substances. 

Water, - 

Iron, 

Glass-crystal, 

Mercury, 

Lead, 

Tin, 

Sulphur, 

Lime, burned, 

9 Water, 10 Lime, 

Sulphuric acid, sp. g. = 1-87058, 

Nitric acid, sp. g. = 1*29895, 

-Alcohol, sp. g. = 0'"' 

Platinum, 

Antimony, 

Zinc, 

Copper, - 

Iron, - 

Glass, - 

Gold, - 

Bismuth, 

Woods in average, 

Sweet Oil, 

Nickel, 

Cobalt, - 

Tellurium, 



SPECIFIC CALORIC OF GASES AT EQUAL 

DENSITY. 
Air, atmospheric, 
Hydrogen, 
Oxygen, 
Nitrogen, 
Carbonic-oxid gas, 
Carbonic acid, - 
Nitro-oxid gas, 
Gas of oils, 
Steam, - 



Specific Caloric. 


32° to 212°. 


32°, 572°. 


1.0000 






0.1105 






0.1929 






0.029 


to 0-033 


0035 


0.02819 


to 0-0293 




0.04755 


to 0-0514 




0.2085 


to 0-188 




0.2169 






0.43912 






0.3346 






0.66139 






0.7 






0.0344 


to 0-0335. 


0-0355 


0.0507 




0-0547 


0.0927 




0-1015 


0.094 


to 0-0949 


0-1013 


0.1098 


to 0-1105 


0-1218 


0.1770 




0-19 


0.0288 






0.0298 






0.48 


to 0-6 




0.30961 






0-1035 






0-1498 






0-0912 






Volume. 


Weight. 


Weight 


air = 1. 


air = 1. 


water = 1 


1-000 


1-000 


0-2669 


0-9033 


12-34 


3-2936 


0-9764 


0-8848 


0-2361 


1-0000 


1-0318 


0'2754 


1-034 


1-0805 


0-2884 


1-2583 


0-828 


0-221 


1-3505 


0-8878 


0-2369 


1-553 


1-5763 


0-4207 


1-96 


3-136 


0-847 



Capacity for Caloric is the relative ability of bodies to retain the specific caloric. 
Capacity for caloric is inverse as the density of the substances. The specific 
caloric multiplied by the atom weight of a substance, gives the constant number 
0*375 (average) which proves that the atoms have equal capacity for caloric in 
all substances. This is a fact with no known reason, but by it valuable results 
may be opened. 

Table of Relative Capacity for Caloric. 



Names. 


Equal 


Equal 


Names. 


Equal 


Equal 


Weights. 


Volume, 


Weights. 


Volume. 


Water 


1-000 


1-000 


Zinc 


0-102 




Copper 


0-114 


1-027 


Tin 


0-060 




Iron 


0-126 


0-993 


Lead 


0*043 


0-487 


Brass 


0-116 


0-971 


Glass 


0187 


0-443 


Gold 


0-050 


0-966 








Silver 


0-082 


0-833 









When the volume diminishes the capacity for caloric will also be diminished 
and thus part of the caloric will profuse the body. A volume of air compressed 
to £ its bulk will fire tinder, which requires a temperature of about 550°. 

21* 



246 Steam. 

_____ 

Steam is the vapour into -which water is converted* by the appli- 
cation of heat. 

Let AB be a cylindrical glass tube in which is fitted a piston a of 
one square inch area; consider this piston to have no friction or 
weigh:, and can be moved steam-tight from A to _ . Let the 
tube be 1723 inches from A to _, the space under the piston a just 
one inch from the bottom being filled with water of 32° _ ah., which 
will be one cubic inch ; weigh the whole apparatus. Xow, place a 
lamp under the tube in a position as represented by the Figure, 
and notice the time, (say 107?, 5m.) The temperature of the water 
will gradually increase, and the piston a maintain a contact with it 
until the water begins to boil, which time is to be carefully noticed; 
now (10fi, lorn.) It will be found that temperature of the water has 
raised from 32° to 212°, which took 10/i, 15m — 10ft 5m = 10 
minutes. 

Let the lamp still remain and the boiling be continued. The 
piston a will now leave the water, and gradually ascend towards B, 
apparently leaving a space between itself and the water, the latter 
will gradually diminish a3 the piston ascends, which indicates that 
steam is gradually formed, and occupies the space between the water 
and the piston, and as the piston has no weight or friction it is 
evident that the density of the steam must be the same as the sur- 
rounding atmosphere. 

But another important faculty of steam and water will now be 
manifest, namely, that the temperature of both will remain the 
same, 212°, as at the boiling point, (10/i lbm,) consequently the heat 
from the lamp which goes into the water and steam is not sensible 
but becomes latent. The water is now getting very low; observe 
carefully the moment when it apparently disappears on the bottom 
of the tube. - - <; now" (llh 10m..) The piston a will be found at 
B, 1700 inches from A, and the time from the boiling-point is 
(117i 10m) — (lO/i lbm) = 55 minutes = 5£ times that occupied to 
raise the water from 32° to 212° = 180° ; hence the quantity of heat 
from the lamp now contained in the steam is 180X5^+180 = 1170° 
of which 180° is sensible and 990 latent. If the water had been en- 
closed in a vessel to prevent evaporation, and the same quantity of 
heat 1170° imparted to it, it would have a temperature of 1202, 
which is about that of metals when red hot in daylight. 

Again to the tube A B, at the time before noticed, viz., Ylh, 10m. 
Take the lamp away, weigh the apparatus, and it will be found the 
same weight as before ; hence, the same quantity of water is still in 
the tube, but in the form of steam. The heat will now radiate from 
the tube, and it will be observed that the piston a gradually 
descends towards A, and the inner surface of the tube will be cov- 
ered with a dew which will soon fall to the bottom as water, but still 
maintain the heat of 212°, until the piston a has fully reached its 
former position at A, when the same quantity of water (one cubic 
inch) will occupy the same space as before the lamp was put under 
it, but with a temperature of 212°. 

The heat required to make steam of one cubic inch of water is able 
to raise 5£+l cubic inches from 32° to 212° ; or steam at 212°, formed 
of one cubic inch of water, can raise 5£ cubic inches of ice cool water 
from 32° to 212°, when mixed together, making 6£ cubic inches. 

E_ect of Steam* 

By the preceding experiment we find that one cubic inch of water 
will be 1700 cubic inches converted into steam ; or one cubic inch of 
water makes one cubic foot of steam 212°, the same density as the 
surrounding atmosphere which is 11| pounds per square inch ; the 
effect of steam in the experiment wa3 consequently a weight of 
F= ll? pounds raised 1700 inches = 112 feet in 55 minutes, or 



W 



Steam. 247 



F, 1*76X148.. . 635 Effect8 _ 
t 55X&> 

See Formula 5, page 148. 

Advantage of Using High Steam. 

Let us now make the same experiment with the tube AB, and load the piston 
with 14$ pounds, which will he a weight F= 29'5 pounds including the atmos- 
phere. Set the lamp under as before, and the experiment is in operation. The 
temperaturo of the water will now not cease to increase when it has attained 
212° ; nor will the piston a begin to raise after 10 minutes as in the former 
experiment; but, when the w.ter has attained 250°, it will cease to increase, the 
piston commence to ascend, and steam to generate. The piston will now only- 
raise 930 inches from A, which will occupy the same time as before. The me- 
chanical effect of the steam is therefore 29*5 pounds raised 930 = 77*5 feet in 55 
minutes, or, 

P= -'^^ = 0-693 Effects, 
which exceeds the former experiment about 9 per cent. 

°'^^- = 0-913 or 100 — 91-3 = 8-6 per cent. 

0-70469 

an advantage of using higher steam. 
This per centage will increase as the steam is used higher. 

Advantage of Using Steam Expansively. 

"We now continue the latter experiment. The piston a stands at 930 inches 
from A ; take the lamp away, and remove the 14| pounds on the piston. The 
steam of 250° in the tube will now raise the piston a to B, at 1700 inches from 
A and the temperature of the steam will decrease from 250° to 212°; conse- 
quently the same steam has produced an additional effect by raising 14£ pounds 
(the pressure of the atmosphere,) 1700 — 930 = 770 inches high = 64'165 feet, 
which for comparison, will here assume to he accomplished in the same time 
55 minutes, we shall then have 

P= li^X6«!5 = 0-29127 Effects 
55X60 

and 0-70469+0-29127 = 0*99596 Effects produced by the same quantity of steam, 

and ^5i^? = 0-646. 100 — 64*6 = 35*3 per cent., 

y*yyyyo 

gained by using the steam high and expansively. 

Advantage taken of the Pressure of the Atmosphere "by 
Vacuum, 

Again to the latter experiment, the piston a stands at B, and the tube is full 
of steam at 212° ; let there now be introduced among the steam 5£ cubic inches 
of ice cool water, (32°), the steam will immediately condense to water, and the 
piston a begin to descend ; finally, between the piston and the bottom of the 
tube will be found 6£ cubic inches of water at 212°, hence the atmospheric 
pressure has reproduced an effect equal to that the steam before expended on 
it, or 0-64363 effects. 

The principal features of the application of steam to produce mechanical 
effects are now illustrated, and we will proceed to give the principal Rules, For- 
mulas, and Tables^ respecting its property. 

If steam is reduced in volume, its density and temperature will increase ; and 
when additional heat is applied to steam its density or volume will increase the 
same as if it was produced direct from water. 



I 



248 




Tabli 


of Props 


tties of Steam. 








ALmosph. 


included. 




Jc 




Atmosphere excluded' ■ 


Tempera- 
ture 


Inches of 


Founds 
per square 


Specific 
gravity, 


Volume 
compared 


Number 

of atmos- 


Inches of 


Pounds per 


of Steam' 


Mercury • 


iucii. 


air = i. 


with water. 


pheres. 


Mercury, 


square inch. 


32° 


0-200 


0-09S 


0-0041 


187407 




—29.79 


—14-60 


40 


0-263 


0-129 


0-0053 


144529 


0-01 


-29-73 


-14-57 


50 


0-375 


0-184 


0-0074 


103350 


0-01 


-29-62 


-14-52 


60 


0-524 


0-257 


0-0102 


75421 


0-02 


-29-47 


-14-44 


70 


0-721 


0-353 


0-0136 


55862 


0-02 


-29-27 


-14-35 


80 


1-000 


0-490 


0-0186 


41031 


0-03 


-29-00 


-14.21 


90 


1-36 


0-666 


0-0250 


30425 


0-05 


-28-63 


-14.03 


100 


1-86 


0-911 


0-0333 


22873 


0-06 


-28-13 


-13.79 


103 


2-04 


1-000 


0-0364 


20958 


0-07 


-27-95 


-13.70 


110 


2-53 


1-240 


0-0458 


16667 


0-08 


-27-46 


-13 46 


120 


3-33 


1-632 


0-0576 


13215 


0-11 


-26-66 


-13*07 


130 


4-34 


2-129 


0-0538 


10328 


0-14 


-25-65 


-12*57 


140 


5-74 


2-813 


0-0960 


7938 


0-19 


-24-25 


-11*89 


145 


6-53 


3-100 


0-108 


7040 


0-22 


-23-46 


-1160 


150 


7-42 


3-636 


0-122 


6243 


0-25 


-22-57 


-1106 


155 


8-40 


4-166 


0-137 


5559 


0-28 


-21-59 


-10-54 


160 


9-46 


4-635 


0-153 


4976 


0-31 


-20-53 


-10»07 


165 


10-68 


5-23 


0-171 


4443 


0-35 


-19-31 


- 9 47 


170 


12-13 


5-94 


0-193 


3943 


0-4 


-17-86 


- 8*76 


175 


13-62 


6-67 


0-215 


3538 


0-45 


-16-37 


- 8*03 


180 


15-15 


7-42 


0-238 


3208 


0-50 


-14-84 


- 7'28 


185 


17-00 


8-33 


0-265 


2879 


0-56 


-12-99 


- 6*37 


190 


19-00 


9-310 


0-294 


2595 


0-63 


-10-99 


- 5*39 


195 


21-22 


10-40 


0-325 


2342 


0-71 


- 8-77 


- 4*30 


200 


23-64 


11-58 


0-36 


2118 


0-79 


- 6-35 


- 3*12 


205 


26-13 


12-80 


0-394 


1932 


0-87 


- 3-86 


- 1*90 


210 


28-84 


14-13 


0-431 


1763 


0-96 


- 1-15 


- 0*57 


211 


29-41 


14-41 


0-440 


1730 


0-98 


+ 0-58 


- 0*29 


212 


30-00 


14-70 


0-448 


1700 


1-00 


+ 0-00 


+ 0*00 


212-8 


30-60 


15- 


0-457 


1669 


1-02 


4- 0-60 


+ 0-30 


214-5 


31-62 


15-5 


0-471 


1618 


1-05 


-f- 1*62 


+ 0'80 


216-3 


32-64 


16- 


0-484 


1573 


1-09 


+• 2-64 


+ 130 


218- 


E3-66 


16-5 


0-497 


1530 


1-12 


+ 3-66 


+ 1*80 


219-6 


34-68 


17- 


0-512 


1488 


1-15 


4- 4-68 


+ 2*30 


221*2 


35-70 


17-5 


0-529 


1440 


1-19 


4- 5-70 


+ 2-8 


222-7 


36-72 


18- 


0-540 


1411 


1-22 


+ 6-72 


+ 3-3 


221-2 


37-74 


18-5 


0-554 


1377 


1-25 


+ 7-74 


+ 3*8 


225-6 


38-76 


19- 


0-567 


1343 


1-29 


+ 8-76 


+ 4-3 


227-1 


39-78 


19-5 


0-581 


1312 


1-33 


+ 9-78 


+ 4-8 


228.5 


40-80 


20- 


0-595 


1281 


1-36 


+10-80 


+ 5-3 


229-9 


41-82 


20-5 


0-608 


1253 


1-40 


+ 11-82 


+ 5*8 


231-2 


42-84 


21- 


0-612 


1225 


1-43 


+12-84 


+ 6-3 


232-5 


43-86 


21-5 


0-636 


1199 


1-46 


+13-86 


+ 6-8 


233-8 


44-88 


22- 


0-05 


1174 


1-50 


+14-88 


+ 7-3 


235-1 


45-90 


22-5 


0-663 


1150 


1-53 


+15-90 


+ 7*8 


236-3 


46-92 


23- 


0-677 


1127 


1-56 


+16-92 


+ 8-3 


237-5 


47-94 


23-5 


0-690 


1105 


1-60 


+17-94 


+ 8-8 


238-7 


48-96 


24- 


0-704 


1084 


1-63 


+18-96 


+ 9-3 


239-9 


49-98 


24-5 


0-717 


1064 


1-67 


+19-98 


+ 9-8 


241- 


51-00 


25- 


0-730 


1044 


1-70 


+21-00 


+10-3 


243 3 


53-04 


26- 


0-756 


1007 


1-77 


+23-04 


+11-3 







TAttLE OP PitOPiRTIES OF Sl'E 


\M. 




2*9 




Atmosph. 


'included. 




k 




Atmosphere excluded* , 


Tempera- 
ture 
of Steam, 


Inches of 
Mercui.v. 


Poumid 

per square 

Inch. 


Specific 
gravity, 
air = 1" 


Volume 
compared 
with water. 


Number 
of atmos. 
pheres. 


Inches of 
Mercury. 


Pounds pei j 
square Inch. 1 


245*5° 


55-08 


27 


0-784 


973 


1-83 


+25-08 


+ 12-3 j 


247'6 


57-12 


28 


0-810 


941 


1-90 


+27-12 


+13-3 


249-6 


59-16 


29 


0-836 


911 


1-97 


+29-16 


+14-3 


251-6 


61-20 


30 


0-863 


883 


2-04 


+31-20 


+ 15-3 


253-6 


63-24 


31 


0-889 


857 


2-11 


+33-24 


+16-3 


255-5 


65-28 


32 


0-915 


833 


2-18 


+35-28 


+ 17-3 


257*3 


67-32 


33 


0-941 


810 


2-24 


+37-32 


+18-3 


259-1 


69-36 


34 


0-968 


788 


2-31 


+39-36 


+19-3 


260-9 


71-40 


35 


0-993 


767 


2-38 


+41-40 


+20-3 


262-6 


73-44 


36 


1-020 


748 


2-45 


+43-44 


+21-3 


264-3 


75-48 


37 


1-045 


729 


2-52 


+45-48 


+22-3 


265-9 


77-52 


38 


1-071 


712 


2-59 


+47-52 


+23-3 


267-5 


79-56 


39 


1-097. 


695 


2-65 


+49-56 


+24-3 


269-1 


81-60 


40 


1-122 


679 


2-72 


+51-60 


+25-3 


270-6 


83-64 


41 


1-148 


664 


2-79 


+53-64 


+26-3 


272-1 


85-68 


42 


1-175 


649 


2S6 


+55-68 


+27-3 


273-6 


87-72 


43 


1-200 


635 


2-92 


+57-72 


+28-3 


275* 


89-76 


44 


1-225 


622 


3-00 


+59-76 


+29-3 


276-4 


91-80 


45 


1-249 


610 


3-06 


+61-80 


+30-3 


277-8 


93-84 


46 


1-275 


598 


3-13 


+63-84 


+31-3 


279-2 


95-88 


47 


1-567 


586 


3-20 


+65-88 


+32-3 


280-5 


97-92 


48 


1-325 


575 


3-26 


+67-92 


+33-3 


281-9 


99-96 


49 


1-351 


564 


3-32 


+69-96 


+34-3 


283-2 


102-0 


50 


1-376 


554 


3-40 


+72-00 


+35-3 


284-4 


104-0 


51 


1-400 


544 


3-47 


+74-00 


+36-3 


285-7 


106-1 


52 


1-426 


534 


3-53 


+76-1 


+37-3 


286-9 


108-1 


53 


1-450 


525 


3-60 


+78-1 


+38-3 


28S-1 


110-2 


54 


1-477 


516 


3-67 


+80-2 


+39-3 


289-3 


112-2 


55 


1-500 


508 


3-74 


+82-2 


+40-3 


290-5 


114-2 


56 


1-523 


500 


3-81 


+84-2 


+41-3 


291-7 


116-3 


57 


1-548 


492 


3-88 


+86-3 


+42-3 | 


292-9 


118-3 


58 


1-575 


484 


3-94 


+ 88-3 


+43-3 


294-2 


120-4 


59 


• 1-598 


477 


4-01 


+90-4 


+44-3 


295-6 


122-4 


60 


1-621 


470 


4-08 


+92-4 


+45-3 


296-9 


124-4 


61 


1-646 


463 


4-15 


+94-4 


+46-3 


298-1 


126-5 


62 


1-671 


456 


4-22 


+96-5 


+47-3 . 


299-2 


128-5 


63 


1-698 


449 


4-28 


+98-5 


+48-3 


300-3 


130-5 


64 


1-719 


443 


4-35 


+100-5 


+49-3 


301-3 


132-6 


65 


1-743 


437 


4-42 


+102-0 


+50-3 


302-4 


134-6 


66 


1-755 


434 


4-49 


+104-6 


+51-3 


303-4 


136-7 


67 


1-794 


425 


4-55 


+106-7 


+52-3 


304-4 


138-7 


68 


1-818 


419 


4-62 


+108-7 


+53-3 


305-4 


140-8 


69 


1-839 


414 


4-69 


+110-8 


+54-3 


306-4 


142-8 


70 


1-868 


408 


4-76 


+112-8 


+55-3 


307-4 


144-8 


71 


1-891 


403 


4-82 


+114-8 


+56-3 


308-4 


146-9 


72 


1-915 


398 


4-89 


+116-9 


+57-3 


309-3 


148-9 


73 


1-938 


3J3 


4-96 


+118-9 


+58-3 


310-3 


151-0 


74 


1-963 


3;8 


5-03 


+121-0 


-r59-3 


311-2 


153-0 


75 


1-991 


383 


5-09 


+123* 


+60-3 


312-2 


155-1 


76 


2-011 


379 


517 


+125-1 +61-3 


313-1 


157-1 


77 


2<t)36 


374 


5-23 


a.127-1 


+62-3 



250 




Table 


of Pr.or-ErTiES of S 


TEAM - . 








Atmosph. 


included 




7c 




1 Atmosphere excluded. 


Tempera- 




, p oun( ) 


Specific 


Volume 


Number 


i 


ture 


Inches of 


os 


gravity, 


Compared 


of atmos- 


; Inches of 


Pounds per 


of Steam. 


Mercury. 


Inch. 


air = 1. 


with water. 


pheres. 


Mercury. 


square inch. 


314-0° 


159-1 


7S 


2-060 


370 


5-30 


+129-1 


+ 63-3 


314-9 


161-2 


79 


2-0S1 


366 


5-37 


+131-2 


+ 64-3 


315-8 


163-2 


80 


2-105 


362 


5-44 


+133-2 


+ 65-3 


316-7 


165-3 


81 


2-128 


358 


5-51 


+135-3 


+ 66-3 


317-6 


167-3 


82 


2-152 


354 


5-57 


+137-3 


+ 67-3 


318-4 


169-3 


83 


2-178 


350 


5-64 


+139-3 


+ 68-3 


319-3 


171*4 


84 


2-203 


346 


5-71 


+141-4 


+ 69-3 


320-1 


173-4 


85 


2-228 


342 


5-78 


+143-4 


+ 70-3 


321-0 


175-5 


86 


2-248 


339 


5-85 


+145-5 


+ 71-3 


321-8 


177-5 


87 


2-275 


335 


5-91 


+147-5 


+ 72-3 


322-6 


179-6 


88 


2-295 


332 


5-98 


+149-6 


+ 73-3 


323-5 


181-6 


89 


2-322 


328 


6-05 


+151-6 


+ 74-3 


324-3 


183-6 


90 


2-343 


325 


6-12 


+153-6 


+ 75-3 


325-1 


185-8 


91 


2-365 


322 


6-19 


+155-6 


+ 76-3 


325-9 


187-8 


92 


2-3S9 


319 


6-26 


+157-8 


+ 77-3 


326-7 


189-8 


93 


2-411 


316 


6-32 


+159-8 


+ 78-3 


327-5 


191*9 


94 


2-435 


313 


6-39 


+161-9 


+ 79-3 


328-2 


193-9 


95 


2-459 


310 


6-46 


+163-9 


+ 80-3 


329-0 


196-0 


96 


2-483 


307 


6-53 


+166-0 


+ 81-3 


329-8 


198-0 


97 


2-505 


304 


6-60 


+168-0 


+ 82-3 


330-5 


200-0 


98 


2-530 


301 


6-66 


+ 170*0 


+ 83-3 


331-3 


202-0 


99 


2-558 


298 


6-73 


+172-0 


+ 84-3 


332-0 


204-0 


100 


2-583 


295 


6-SO 


+174-0 


+ 85-3 


335-8 


214-2 


105 


2-703 


282 


7-13 


+194-2 


+ 90-3 


339-2 


224-4 


110 


2-815 


271 


7-47 


+194-4 


+ 95-3 


342-7 


234-6 


115 


2-947 


259 


7-82 


+204-6 


-1100-3 


345-8 


244-8 


120 


3-036 


251 


8-15 


+214-8 


4105-3 


349-1 


255-0 


125 


3-178 


240 


8-5 


+225-0 


+ 110-3 


352-1 


265-2 


130 


3-270 


233 


8-83 


+235-2 


+115-3 


355-0 


275-4 


135 


3-405 


224 


9-16 


+245-4 


+ 120-3 


357-9 


285-6 


140 


3-497 


218 


9-51 


4-255*6 


+ 125-3 


360-6 


295-8 


145 


3-626 


210 


9-83 


+265-8 


+ 130-3 


363-4 


306-0 


150 


3-712 


205 


10-2 


+276-0 


+ 135-3 


368-7 


326-4 


160 


3-941 


193 


10-9 


+296-4 


+ 145-3 


373-6 


346-8 


170 


4-028 


183 


11-5 


+316-8 


+ 155-3 


378-4 


867-2 


180 


4-375 


174 


12-2 


+337-2 


+165-3 


382-9 


387-6 


190 


4-585 


166 


12-9 


+357-6 


+175-3 


387-3 


408-0 


200 


4-82 


158 


13-6 


+378-0 


+ 185-3 


403-8 


509- 


250 


5-90 


129 


17-0 


+479- 


+235-3 


420-3 


612* 


300 


7-00 


109 


20-4 


+582- 


+285-3 


435-0 


714- 


350 


8-00 


95 


23-8 


+684- 


+345-3 


446-5 


816- 


400 


8-95 


85 


27-2 


+786- 


+385-3 


471-3 


1019- 


500 


10-9 


70 


34-0 


+989- 


+485-3 


487-0 


1223- 


600 


12-8 


59 


40-S 


+1193- 


+585-3 


519- 


1631- 


800 


15-7 


4S 


54-4 


+1601- 


+785-3 


548. 


2038- 


1000 


19-7 


38 


68 


+2008- 


+985-3 





Steam. 251 

Letters denote. 
JF=* force of the steam, or pressure per square inch in pounds. 
J= inches of Mercury that balances the steam. 
T= temperature of the steam in degrees of Fahrenheit's Thermometer. 

Formulas for Steam above 212°. 

j =(ii+°- 584 y % 

T=( yT— 0-52)202, 3. 

>erature of a quantity of steam is foun 
pounds per square inch ? 

(275 \« 

2^2+0-52 1 = 44*3 pounds per square inch. 



Example 1. The temperature of a quantity of steam is found to be 275°. 
Required the density in pounds per square inch 1 

Tib 



275 
By logarithms, 202 + °' 52 = 1 ' 881 ' 

log.1-881 = 0-274389 

6 

log.44-3 = 1-646334 or 44*3 pound per square inch. 

The properties of steam are calculated and contained in the accompanying 
Table, as noted on the top. The two last columns contain the inches of mercury 
and pressure per square inch commonly expressed in practice ; it is O at the 
temperature 212°, and below that temperature it is negative, which denotes so 
much vacuum. If the temperature in a condenser is 120, the vacuum is 13*07 
pounds. 

To Find tlie Weight of Steam, 

RULE. Multiply the specific gravity of the steam by the weight of one cubic 
foot of air = 0-07529, and the product is the weight per cubic foot of the steam 
in pounds. 

To Find the Quantity of Water ofwTiich a given quantity of Steam lias been, or can 
be produced. 
RULE. Divide the cubic contents of the steam by the volume k in the Table, 
and the quotient is the cubic content of the water. 

Force or Feed Pumps, 
Letters denote. 

d s Z stroke!^ } of the force P um P> sin S le ™ tin S' 
£ = stroke } of tne steam cylinder piston, in inches, double acting. 
lc = volume given in the Table at the given pressure of steam. 
The stroke of the steam piston is only that under which steam is admitted to 
the cylinder. 



l =">\/li 



'kd* t 



d = 2L 
Slip water included. 



252 Air-Pump. 

Example. Required the diameter of a force-pump having the same stroke as 
the cylinder piston s = 38 inches, diameter of cylinder D = 30 inches, the steam 
is cut off at i the stroke, and the steam pressure -f- 50 pounds per square inch ? 
Here k = 437, and S = 19 inches, because steam is cut off at £ the stroke. 



*== 8X80 y/jj^,- 2-03 inches. 



To find the Quantity of Condensing Water, 
Letters denote, 
q = condensing -water of temperature t, in cubic feet. 
Q = steam of temperature T, in cubic feet. 
k — volume in the Table. 
t f = temperature in the condenser when the steam and water are mixed. 

„_ l-4g(990+!r— Q . 

2 k {t'-t) » " " " " ■ ' * 

Dimensions of the Air Pump* 

, = ?troke er } of the *** P um P> sin S le acting. 
S == stroke ^ \ °* ^ ie s ^ eam cylinder, double acting. 

Assume ^ = 100°, and £ = 50°, we shall have, 



d = 0-S26ZK /gwo±2? t 

t-wuiej»*P <0 + 2 '>, 




single acting air-pumps. 

9, 

10, 
double acting air pumps. 

11. 

Example. A single acting air-pump is to be constructed for an engine 
D = 38 inches, S = 45 inches stroke of the cylinder ; the stroke of the air- 
pump can be 32 inches, and the exhaust steam is 261°. Required the diameter 
of the air-pump ? k = 767. 

d = 0-326.X384 /-^ 3 2 2- = 18-25 inches. 

j$3r*Slip water included. Tand k must be taken for the exhaust steam, as the 
steam may have had worked expansively ; the area of the foot valve must be 
calculated from the following formulas. 

Foot Valve in the Air Pump. 

To render an air-pump to work well, and with the greatest advantage, it is 
necessary to pay particular attention to the following formulas. The force by 
which the water is driven from the condenser through the footvalve into the 
air-pump is limited by the pressure in the condenser; this pressure is the 
vacuum subtracted from 14'7 pounds ; it is noted in the third column where 
the temperature in the condenser is opposite, in the first column. Every 
pound of this pressure per square inch balances a column of water 27 inches 
high, which is the head that presses the water from the condenser. 



Am-Ptnir. 

Letters denote. 
gj = area of the air-pump piston. 

a b area of the foot-valve, or bucket-valve. 
33 = diameter of the air-pump-piston. 

to = diameter of the foot-valve, when round. 
S = stroke of air-pump piston, in feet. 
JJjJ = pressure in the condenser at the temperature T. 

n = number of strokes of the air-pump piston per minute. 



_ D-S??( c 90-f J) 
~~ 23000 mfcy'p"' 
m = 0-6 to 0*8 



a = 



S = 



100 vW 

1003 yg 

lOOa^/p 

&3 ' 



12, 
13, 
14, 



== "loW r, 



S = 



lOOWjp 



tt^lOOWg 
23* S ' 



15, 

16, 
17. 



Example. The area gj of an air-pump-piston is 2*35 square feet, stroke of 
piston 5b == 3 ' 6 feet, to make n = 40 strokes per minute, and the pressure to be 
J3 = 3-2 pounds. Kequired the area of the foot-valve. 

„ 2-35X3-6X40 

a = • mn /-o^- "= 1,8 ° S( iuare feet. 



To Find the Velocity and Quantity of the Injection Water 
through the Adjustage into the Condenser. 

Letters denote, 
v = velocity in feet per second. 

h = head of the press water; -f- "when above, and — below the adjustage. 
F= vacuum, noted — or negative in the last column, but is positive in the 
formulas. 
q — quantity of water discharged in cubic feet, per second. 
a = area of all the holes in the adjustage in square feet. 
d= diameter") -.* . ... . . „ . 

L = length f injection pipe, in feet. 

n = double strokes of cylinder-piston, or revolutions per minute. 

A, D, and S, dimensions of the steam cylinder, in feet. 

T = temperature, and k — volume coefficient of the exhaust steam. 



fl = 5~3F±p 18, 

t>-8,/2F±A 19, 



nSD ^MO+T) 20 
q 55k ' ' 



22 



q = 5a ^2F+h, 



21, 



d =°™\/Mi>> 22 > 



nS D 2 (940+ T) 
?15kV2F+h ' 



23, 



254 Steam. 

Example. Required the diameter of an injection pipe L = 10 feet long, 
which shall supply q = 1-3 cubic feet of water per second into a vacuum of 12 
pounds per square inch, the head of press water h = 2 feet ? 

d = 0-35 A 5 /. 1Q Xl-3_ = . 3055 feet = 3 i j. inches. 
\/ 2X12+2 lt> 

Area of Steam Passages* 

A = area of the steam pipe, sq. in. 

A = area of the cylinder piston, sq. in. 

d = diameter of the pipe, in inches. 

D = diameter, S = stroke of cylinder, in inches. 

a - 86000' d ~ 186-' 24 ' 25 ' 

Example. Required the diameter of a steam-pipe for a cylinder D = 40 
inches. Stroke of piston S = 48 inches, and n = 38 revolutions per minute ? 

cZ = W^X38 = 9-2 inches, nearly. 
186 

^Steam Ports to the Cylinder. 

ASn OR 

a = ■ . • • • • *o. 

a 30600' ' 

Safety Valve* 

Three-fourths of the fire grate in square feet is a good proportion for the 
safety valve in square inches. 

Notation of Letters corresponds with Figure 3, Plate VIII. 
a = area of safety valve in square inches, 
P= pressure per square inch in the boiler 
W= weight on the safety valve lever Vin pounds. 

Q = weight of the safety valve and lever 
I = lever for W ^ 
e = " aP fin inches, 

x= « Q) 
Balance the lever over a sharp edge, and the centre of gravity Q is found 
measure the distance x from the fulcrum (7. 



> 



aPe = WZ+Qr 27, 



ae 



i 



aPe — Q x 

W ■ 



30, 



Example. Area of the safety valve a =9 square inches, e = 4£ inches, 
W= 50 pounds, weight of the lever and safety valve Q — 15 pounds, and x = 17 
inches Required at what distances I, V and I" will the weight W indicate pres- 
sures of P = 30, P' = 40, and P" = 50 pounds ? 



Z = 



9X30X4-5-15X17 = 2Q . 2 



50 



from the fulcrum Cthe weight TTwill indicate P= 30 pounds. 

V = 37-9 inches, when P f = 40 pounds. 

I" = 45-8 " " P" = 50 " 

and thus the lever can be graduated. 



Expansion of Steam. 256 



EXPANSION OF STEAM. 

In order to save steam, or more correctly to employ its effect to a higher 
degree, the admittance of steam to the cylinder is shut off when the piston 
has moved a part of the stroke ; from the cut-off point the steam acts ex- 
pansively with a decreased pressure on the piston, as represented by the 
accompanying figure. 

Let the steam be cut off at I of the stroke, and 
Aa represent the total pressure, say 20 pounds 
per square inch which will continue to the point 
E where the admittance of steam is shut off at 
one-third the stroke S. The steam Aa eE, is now 
acting expansively on the piston, and the pres- 
sure decreases as the volume increases, when the 
piston has attained Cc or two-thirds of S, the 
pressure C'c= 10 pounds, only half the pressure 
Aa=20 because the volume Aa eE is only half of 
Aa cC, and so on until the piston has attained B b 
the pressure Bb=% X 20 =6-66 pounds. 

The mean pressure, or the effectual pressure, 
throughout the stroke, will be about 13-33 pounds 
per square inch, or 66 per cent., but the quantity 
of steam used is only 33 per cent., hence 33 per 
ii^v cent, is gained by using the steam expansively. 
I = part of the stroke S in feet, at which the steam is cut off. 
P— pressure per square inch under full admittance of steam. 
IT = mean pressure per square inch throughout the stroke S. 
/=mean pressure per square inch during the expansion, which in 
double expansion cylinder engines will be the average pressure per 
square inch on the large piston A. 
p = end pressure per square inch after expansion. 
S = stroke of the cylinder Piston in feet. 

r== _ (2 . 3 i og .s+i) *— inn F =s- 

The following Tables are calculated from these formulas. 
Example 1. Required from the Table I. the mean pressure Ffor P=32 
lbs. at five-eights expansion. 

Add { F °* 30 lbs. £=22-252 j from ^ ^^ L 

Mean pressure of 32 lbs. F=23-735 the answer. 
Example 2. Required from Table II. the mean pressure /, per square 
inch during the expansion, or on the large piston A in double cylinder 
engines, when the initial pressure P=75 lbs. and under two-thirds ex- 
pansion 1 /=40-75 Table II. 

Example 3. Required the mean pressure f=1 for an initial pressure 
P=43 lbs. under % expansion 1 ? 

For P = 40 lbs. /= 18-48 ) T ., „ 
P = 30 or 3 lbs. / = 1-38 | laDle n * 

P = 43 lbs. /= 19-86 the answer. 

The effect gained or fuel saved by expansion and high steam is calculated 

from the following formulae, in which it is supposed as a unit the work 

of an engine with P=30 pounds per square inch, or an indicated pressure 

of 15 lbs. without expansion. 

c = per cent on 100, of effect gained or fuel saved. 

IP 26490 

For expansion c = 100 (1—- ). For high steam c = 100 (1 ). 

or kP 

The following Table III. is calculated from these formulae, in which 
the first line from 30 contains the economy per cent, from expansion 
alone, and the column o contains the economy per cent, from high steam 
above P=30 lbs. The balance of the table contains the jointed economy 
of expansion and high steam. Required the jointed economy of P=90 
lbs. under £ expansion 1 ? 60*5 per cent, the answer. 



256 



Expansion Table I. Mean Pressure F. 



Prww, 

P. 



Grade of JExpan§ion# 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

IT 

18 

19 

20 

21 

22 

23 

2\ 

2b 

2Q 

28 

30 

35 

40 

45 

50 

55 

60 

65 

70 

75 

80 

85 

90 

95 

100 

105 

110 

115 

125 

140 

i 150 

j 200 

I 250 

| 300 

i 



1-9275 

2-5912 
3-855 3 
4-8185 
5-7822 
B-746S 
7-7106 
5-6732 

i; n , 
1 1 ■ 5 6 5 
12-528 

13*492 
14-456 
15-420 
16-383 
17-347 

15-3.11 

19-275 



- 

12-1 : 

14-933 

1 

17-71 



;■-■ : 

56751 



25-057 
26-9S5 
25-912 
33' 731 

48-187 

5 3- Li 05 

17-822 

. 41 

72-275 
77-091 
51-914 

16-371 
101-18 

I : 

14 ■; i 
192-75 

289-12 



24-26C 

42- 



12-862 
13-781 
14-700 

1 6 - 5 3 7 
17445 
15-375 
19*293 

21-131 

23-887 
25-714 
27-562 
32*156 

- : 
41-: 





45 137 


51-333 






55-122 




59*715 


65-333 








".' 






7 :■:;■? 








87-273 




91-871 












I 








128-62 




137-51 




183-75 


233-33 


229-68 


279-99 


275-62 



0-5765 0-7417 
1-4835 

: 2-9670 

4-2325 3-7055 

. 4-4c .2 

5-9255 5-2419 

5-7721 5-9346 

7-6185 6-6753 
7-4170 

?-3115 5-1597 

11-004 |! 9-6421 

111-126 

13-531 

14-535 
15-57 6 
15-623 16-315 

2'"-316 17-502 

: 

22-252 
. 27 2: -1 
> : ? . • 5 : 29-670 

33-375 
42-325 37-067 
46-557 40-775 

44-520 
:" __ 48-228 

255 12 41 
S3-4S7 56-127 
67*72( " 



II- 
12-1 






2-0873 

3-4955 
4-1946 

5-5928 

6-2919 

9-7:^74 

11-185 
11-884 

13-153 

: 

14-501 
17-477 

27-964 

38*451 
41-946 

45-441 
481 37 



1*7895 
2-9825 

4-1755 

4-7720 

6-5615 
7-158 ■ 
7*7545 

11-333 . 
11-930 
2-52C 
13-123 
13-721 

15-509 

29-825 
32-807 

35-790 
38-772 





_ 


: 


% :_" 


: r 1^1 


Jt " - : 


93120 








: : "- ; ; 


102-81 


: 


148-35 


211-62 


155-43 


253-95 


222-52 



0*7697 

1-1546 
1-5395 

1-9240 

2-6946 
3-0754 
34632 
3-5450 
4-2333 
4-6186 

5-3882 

6-9274 
7-3122 
7-6970 

8-8516 

9-436.5 

9-8978 
10-775 

11-546 

13-470 
15-395 

19-243 

21-167 

24-024 
28-626 

£2-714 

36-554 
38-480 




Expansion Tabli 


II. for Double Cylinder 


Expansion Engines. 257 


Mean Pressure /'during the Expansion. 




Pivs.lJ 1 


i 


3 


A- £ 


2. 3. 


7 


P. (J * 


3 


8 




2 






3 4 


^ 


30 


28-549 


24 


~~~23 r 50 


"io 7 ^ 


lv 


'•60* 


16-31 


13-86 


8-9097 


35 


33-308 


28 


27-41 


24-25 


20-54 


19-02 


16-17 


10-394 


40 


38-066 


32 


31-83 


27*72 


23-47 


21-73 


18-48 


11-879 


45 


42-824 


36 


35-25 


31-18 


26-40 


24-46 


20-79 


13-364 


50 


47*582 


40 


39-16 


34-65 


29-33 


27-16 


23-10 


14-849 


55 


52-340 


44 


43-08 


38-11 


32-24 


30-17 


25-41 


16-334 


60 


57*098 


48 


47-00 


41-58 


35-20 


32-62 


27-72 


17-819 


65 


61-853 


52 


50-91 


45-04 


38-14 


35-33 


30-03 


19-303 


70 


66-616 


56 


54-83 


48-51 


41-07 


38-04 


32-34 


20-788 


75 


71-371 


60 


58-75 


51-90 


44-00 


40-75 


34-65 


22-263 


80 


76-128 


64 


62-66 


55-44 


46-94 


43-47 


36-96 


23-758 


85 


80-885 


68 


66-38 


58-90 


49-87 


46-19 


39-27 


25-243 


90 


86-448 


72 


70-50 


62-37 


52-80 


48-93 


41-58 


26-729 


95 


90-391 


76 


74-41 


65-73 


55-73 


51-62 


43-89 


28-213 


100 


95-160 


80 


78-33 


69-30 


58-66 


54-33 


46-20 


29-699 


105 


99-910 


84 


82-24 


72-76 


61-57 


57-33 


48-51 


31-183 


110 


104-68 


88 


86-16 


76-23 


64-48 


60-35 


50-82 


32-669 


115 


109-40 


92 


90-08 


79-69 


67-44 


62-79 


53-13 


34153 


125 


118-95 


100 


97-91 


97-02 


73-34 


67-95 


57-75 


37-122 


140 


133-23 


112 


109-6 


97-02 


82-14 


76-08 


64-68 


41-576 


150 


142-74 


120 


117-5 


103.9 


88-00 


81-50 


69-30 


44-548 


200 


190-32 


160 


156-6 138.6 


117-3 


108-6 


92-40 


59-398 


250 


237-07 


200 


195-7 173.2 


146-6 


135.8 


115-5 


74-247 


300 


283-16 


240 


235-0 1 207.9 | 


176-0 


163.1 


138-6 


89-097 


Table III. Economy of Expansion and liigli Steam 


i. 


Fuel saved or effect gained per cent. 




Pres.: 





i | 


i 


t 


l 


4 i 2 

5 ! 3 


3 

5 


i 


30 





"T2 


29-5 


32 


41 


49-3 


52 


58 


67-5 


35 


1-6 


13-6 


31 


33-6 


42-6 


51 


53-6 


59-6 


69-1 


40 


2-5 


14-5 


32 


34-5 


43-5 


51-8 


54-5 


60-5 


70 


45 


3-4 


15-4 


33 


35-4 


44-4 


52-7 


55-4 


61-4 


71 


50 


4.3 


16-3 


33-8 


363 


45-3 


53-6 


56-3 


62-3 


71-8 


55 


5-2 


17*2 


34-7 


37-2 


46-2 


54-5 


57*2 


63-2 


72-7 


60 


6 


18 


35-7 


38 


47- 


55-3 


58 


64 


73-5 


65 


6-7 


18-7 


36.2 


38-7 


47-7 


56 


58-7 


64-7 


74-2 


70 


7-3 


19.3 


36.8 


39-3 


48-3 


56-6 


59-3 


65-3 


74-8 


75 


7-8 


19-8 


37-3 


39-8 


48-8 


57-1 


59-8 


65-8 


75-3 


80 


8-5 


20-5 


38 


40-5 


49-5 


57-8 


60-5 


66-5 


76 


85 


9 


21 


38-5 


41 


50 


58-3 


61 


67 


76-5 


90 


9-5 


21-5 


39 


41-5 


50-5 


58-8 


61-5 


67-5 


77 


95 


10 


22 


39-5 


42 


51 


59-3 


62 


68 


77-5 


100 


10-4 


22-4 


40 


42.4 


51.4 


59-7 


62-4 


68-4 


78 


105 


10-7 


22-7 


40-2 


42-7 


51-7 


60- 


62-7 


68-7 


78-2 


115 


11 


23 


40-5 


43 


52 


60-3 


63 


69 


78-5 


125 


11-7 


23-7 


41-2 


43-7 


52-7 


61 


63-7 


69-7 


79-2 


150 


14 


26 


43-5 


46 


55 


63-3 


66 


72 


81-5 


200 


16 


28 


45-5 


48 


57 


65-3 


68 


74 


83-5 


250 


17-7 


29-7 


46-2 


49-7 


58-7 


67 


69-7 


75-7 


85*2 


300 


19 


31 


48-5 

! 


51 


60 


68-3 


71 


77 


86.5 



22* 



258 






Con 


sumption of Fuel 














Table IV. 










Consumption 


of Coal 


in pounds per horse power per h 


our, 






G 


rade of .Expansion. 








PrHS. 
P. o 


! i 


1 * 


! t 


* 


1 * 


i 1 


1 


1 I 

lbs, 


lbsT 


lbs, 


lbs, 


lbs, 


lbs, 


lbs, 


lbs, 


lbs, 


lbs, 


30 


5-6 


4-93 


3-95 


3-81 


3-30 


2-84 


2-69 


2-35 


1-82 


35 


5-51 


4-84 


3-86 


3 72 


3-21 


2-74 


2-60 


2-26 


1-73 


40 


5-46 


4-79 


3-81 


3-67 


3-16 


2-70 


2-55 


2-21 


1-68 


45 


5-41 


4-73 


3-75 


3-62 


3-11 


2-65 


2-50 


2-16 


1-62 


50 


5-36 


4-68 


3-71 


3-57 


3-06 


2-60 


2-45 


211 


1-58 


55 


5-31 


4-63 


3-66 


3-51 


3-01 


2-55 


2-40 


2-06 


1-53 


60 


5-26 


4-59 


3-60 


3-47 


2-97 


2-50 


2-35 


2-02 


1-49 


65 


5-20 


4-55 


3-57 


3-43 


2-93 


2-46 


2-31 


1-98 


1-45 


70 


5-19 


4*52 


3-54 


3-40 


2-90 


2-43 


2-28 


1-94 


1-41 


75 


5-16 


4.49 


3-51 


3-37 


2-87 


2-40 


2-25 


1-91 


1-39 


80 


5-12 


4-45 


3-47 


3-33 


2-83 


2-36 


2-21 


1-88 


1-35 


85 


5-09 


4-42 


3-44 


3-30 


2-80 


2-33 


2-18 


1-85 


1-32 


90 


5-07 


4 "3 9 


3-41 


3-28 


2-77 


2-31 


2-16 


1-82 


1-29 


95 


5-04 


4-37 


339 


3-25 


2-74 


2-28 


2-13 


1-79 


1-26 


100 


5-01 


4-34 


3-36 


3-23 


2-72 


2-26 


210 


1-77 


1-23 


105 


5-00 


4-32 


3-35 


3-21 


2-70 


2-24 


2-09 


1-75 


1-22 


115 


4-9S 


4-31 


3-33 


3-19 


2-69 


2-22 


2-07 


1-73 


1-20 


125 


494 


4-27 


3-29 


3-15 


2-65 


2-19 


2-03 


1-70 


1-17 


150 


4-31 


4-14 


3-16 


3-02 


2-52 


2-05 


1-90 


1-57 


1-04 


200 


4-70 


4-03 


3-05 


2-91 


2 41 


1-94 


1-79 


1-46 


0-92 


250 


4-60 


393 


3-01 


2-81 


231 


1-85 


1-70 


1-36 


0-83 


300 


4-54 


3-S7 


2-89 


2-75 


2«24 


1-78 


1-62 


1-29 


0-75 








Ta 


ble V. 










Cons 


timptio 


n of Co 


al in to 


ns per 


100 hor 


ses in i 


14: hour 


s. 


Pres. 
P. o 


1 * 


1 
3 


t 


h 


S 


2 
3 


1 


i 


lbs. I 


tons 


tons, 


tons, 


tons, 


tons, 


tons, 


tons, 


tons, 


tons, 


30 


6-00 


5-29 


4-23 


4-09 


3-54 


3-04 


2-88 


2-52 


1-95 


35 


5-90 


5-19 


4-13 


3-99 


3-44 


2-94 


279 


2-42 


1-86 


40 


5-85 


513 


4-08 


3-93 


3-39 


2-90 


273 


2 37 


1-80 


45 


5-80 


5-07 


4-02 


3-88 


3-34 


2-84 


2-68 


2-31 


1-73 


50 


5-75 


5 01 


3*97 


3-83 


3-28 


2-79 


2-63 


2-26 


1-69 


55! 


5-70 


4-96 


3-92 


3-77 


3-22 


2-73 


2-57 


2-21 


1-64 


60 


5-64 


4-92 


3-87 


3-72 


3-18 


2-68 


2-52 


217 


1-60 


65! 


5-58 


4-88 


3-82 


3-68 


3-14 


2-63 


2-48 


2-12 


1-55 


70; 


5-56 


4-84 


3-79 


3-64 


3-11 


2-60 


2-44 


2-08 


1-51 


75; 


5-53 


481 


3-76 


3*61 


3-07 


257 


2-41 


2-05 


1-49 


80 


5-49 


4-77 


3-72 


3«57 


3-03 


2-53 


2-37 


2-01 


1-44 


85! 


5-46 


4*74 


369 


3-54 


3-00 


2-50 


2-33 


1-98 


1-41 


90 


5-43 


4-70 


3-66 


3-51 


3-97 


2-47 


2-31 


1-95 


1-38 


95 


5-40 


4-68 


363 


3*48 


2-94 


2-44 


2-28 


1-92 


1-35 


100 


5-37 


4-65 


3-60 


3-46 


2-91 


2-42 


2-26 


1-90 


132 


105 


5-36 


463 


3-59 


3-44 


2-89 


2-40 


224 


1-88 


1-31 


115 


5-34 


4-61 


3-57 


3-42 


2-88 


2-38 


2-22 


1-85 


1-29 


125 


5-30 


4-58 


3-53 


3-38 


2-84 


2-34 


2-18 


1-82 


1-25 


150 


5-16 


4-44 


3.39 


3-34 


2-81 


2-30 


2-04 


1-68 


111 


200 


5-04 


4-32 


3-27 


312 


2-59 


219 


1-92 


1-56 


099 


250, 


4-93 


4-21 


3-22 


3*01 


2-47 


2-09 


1-82 


1-46 


0-89 


300 


4-87 


4-15 


3-10 


2 95 


2-40 


2-01 


1-74 


138 


0-83 



Superheating. 



259 



SUPERHEATED STEAM. 

The Author's experience in superheated steam has been sufficient to 
convince him of its great importance. It appears that in order to utilize 
the maximum effect of steam or at least to attain the maximum quality 
of expansion, it is not necessary to overheat it after a pure steam is 
formed, that is, when all the small particles and bubbles of water are 
evaporated. Water which accompanies the steam in such a form has 
the same temperature as that due to the surrounding steam pressure, 
prevents it to vaporise ; but when it passes through the superheating 
apparatus the temperature is greatly increased while the pressure re- 
mains the same because it being in connection with the steamroom in 
the boiler allows the water to vaporise and a pure steam may be formed. 

If steam with particles of water is admitted into the cylinder part of 
the stroke and then allowed to expand, it is generally found that the end 
pressure does not come up to that due by theory, from which it has been 
pronounced that the expansive quality of steam' does not follow that of a 
perfect gas, and that steam has condensed during the stroke ; but if we 
knew the cubic containt of all the particles of water and subtracted that 
from the cubic containt of the steam it might be found that its expansive 
quality is not so far from that of a perfect gas. It appears also that the 
expansive quality is diminished by overheating pure steam. 

The small particles of water contain a great deal more caloric per 
volume than the surrounding steam, consequently when admitted into 
the condenser a good vacuum cannot be formed so well as with pure 
steam. It is therefore of great importance to pay particular attention to 
the superheating of steam, otherwise economy by expansion will not be 
realized to the extent herein given by formulas and tables. It is also of 
great importance for expansion that the piston and steam valves are per- 
fectly tight. 

SUPEEHEATING APPAEATUS. 

The accompanying figure represents a 
superheating apparatus such as the Au- 
thor has built it in Russia, and is found 
to answer exceedingly well. The figure 
is a section of the forend of an ordinary 
tubular boiler with steamdrum and up- 
take. The chimney is made a great deal 
wider in the steamdrum and contracted 
to the usual size at e, of 0-16 times the 
area of the firegrate ; if a strong fan blast 
is app lied it may be better to contract it 
to 0-11 \ — \ . In the inside of the chimney 
are placed a number of copper tubes a, 
a, b, b, with flanges screwed to the side ; 
the area of these tubes should be about 
four times that of the steampipe c. In 
the steamdrum is riveted steamtight a 
conical plate d, d, so that the steam can- 
not pass to the top without passing the 
superheating pipes. This superheating 
apparatus is in successful operation in 
three first class passenger steamers on 
the River Volga in Russia, each of 500 
actual horses, and one in a steamer of 
100 actual horses on the Black Sea. 

rThe steamdrum can be placed around 
•p.. r \ the chimnev separately from the boiler 
unu LC& f and the steam led either above or below 
x-- the plate d } d, by pipes from the steam- 

as may suit the circumstances, 
superheating apparatus may also be well suited for locomotives. 




room, 
This 



2S0 



Nominal Horsepower of Condensing Engines. 



Diam 


Stroke 


of Cylind 


er Piston S in feet, 


D 

ia. 


V 
H 


11' 3"!'' 6" 1' 9"| 2* 


;2' 3"j|2' 6" 2' 9" 3' 3' 6 


", 4' 4' 6 


"1 5' 


hTi 11 


H 


H 


1 H 


H 


1 H 


1 ll 


H 


~H 


1 H 


1 H 


6 


3-6 


3-88 4-12 


4-32 


4-53' 4-72 


4-8S 


I 5-041 5-19 5-4' 


■ 5-71 5-94' 6-16 


7 


4-9 


5-27; 5*61 


5-9C 


6-17 6-4^ 


6-6c 


1 6-86 7*07 


• 7-44 7-78j 8-10' 8-38 


8 


Q-4 


6-90 7*32 


7-71 


8-06' 8-39 


8-65 


i 8-96 9-2^ 


9-72 10-11 10-6 11-0 


9 


8-1 


8-72 9-27 


9-75 


10-1 


> 10-6 


11-C 


| 11*1 


! 11-7 


12' 


) 12-9 13-4 13-9 


10 


10 


10-8 11'4 


12-0 


12-( 


> 13'1 


13-6 


14-( 


) 14-4 


15-3 


I 15-9 U'l 


> 17-1 


11 


12-1 


13-0 13-9 


14-6 


J 15-5 


> 15-8 


16-4 


! 16i 


17-4 


- 18-i 


19-2 20-( 


) 20-7 


12 


14-4 


15*5 16*5 


17'4 


18-1 


.| 18-9 


19-5 


! 20-2 


20-S 


21-c 


22-9 23-S 


24-6 


13 


16-9 


18-2 19-3 


20-3 


21-2 


22-1 


22-9 


23-7 


24-4 


25-t 


26-8 27*£ 


28-9 


11 


19-6 


2M| 22-4 


23-6 


24-7 


125-7 


266 


|27-4 


28-3 


29-7 


31-lj 32-4 


33-5 


15 


2-25 


24-2 25-8 


27-1 


28-2 


29-5 


30-5 


31-5 


32-4 


34-1 


35-7 37-1 


38-5 


16 


25^6 


27-4 29-3 


30-8 


32-2 


33-5 


34-7 


35-8 


37-0 


38-9 


40-( 


> 42-2 


43-8 


17 


28-9 


311 


33-1 


34-8 


36-4 


37-9 


39-2 


40-5 


41-7 


43-9 


45-c 


I 47-7 


49-4 


18 


32-4 


34-9 


37*1 


39-0 


40-8 


42-5 


44-0 


45-4 


46-8 


49-2 


51-4 


63-5 


55-4 


19 


36-1 


3S-9 


| 41-3 


43-5 


45-5 


47-3 


49-0 


50-6 


52-1 


54-8 


57-3 


59-6 


61-7 


20 


40-0 


43-1 


45*8 


48-2 


50-4 


52-4 


54-3 


56-0 


57-7 


60-7 


63-5 


66-0 


68-4 


21 


44-1 


47'5 


505 


53 - l 


^5-6 


57-8 


59-8 


61-7 


63-6 


67-0 


70-0 


72-8 


75.4 


22 


4:8-4 


52-1 


55*4 


5S-3 


61-0 


63-4 


65-6 


61-8 


69-S 


73-5 


76-8 


80-0 


82-8 


23 ! 


52-9 


57-0 


60-5 


63-7 


66-7 


69-3 


71-8 


74-1 


76-3 


80-3 


84-0 


87-4 


90-5 


241 


57*6 


62-0 


65-9 


69-4 


72-6 


75-5 


78-1 


80-7 


83-1 


87-4 


91-5 


95-2 


98-6 


25 


62-5 


67-3 


71-5 


75*3 


78-7 


81-9 


S4-S 


87*5 


90-2 


94-8 


99-2 


103 


107 


26\ 


67-6 


72-8 


77-3 


81-5 


85-2 


88-6 


91-7 


94-7 


97-5 


102 


107 


111 


116 


27\ 


72-9 


78-5 


83-5 


87-8 


91-9 


95-6 


99-0 


102 


105 


111 


116 


120 


125 


28- 


78-4 


84-4 


S9-8 


94-5 


98-8 


102 


106 


110 


113 


119 


124 


129 


134 


29; 


84.1 


90-5 


96-2 


101 


106 


110 


114 


118 


121 


128 


133 


139 


144 


SO 


90-0 


96-9 


103 


10S 


113 


118 


122 


126 


130 


137 


143 


149 


154 


31 


96-1 


103 


110 


116 


121 


126 


130 


134 


139 


146 


153 


159 


164 


32 


102 


110 


117 


123 


129 


134 


138 


143 


148 


155 


163 


170 


175 


33 


109 


117 


124 


131 


137 


142 


147 


152 


157 


165 


173 


180 


186 


34 


115 


124 


132 


139 


145 


151 


157 


162 


167 


175 


1S3 


190 


198 


35 


122 


132 


140 


148 


154 


160 


166 


172 


177 


186 


194 


202 


210 


38 


129 


140 


148 


156 


163 


170 


176 


182 


187 


197] 


205 


214 


222 


37 


137 


147 


156 


165 


172 


ISO 


1S6 


192 


198 


208 


217 


226 


234 


38 


144 


155 


165 


174 


182 


190 


196 


202 


209' 


218] 


229 


238 


247 


39 


152 


164 


174 


183 


192 


200 


206 


213 


220| 


231 


241 


251 


260 


40 


160 


172 


183 


193 


202 


210 


217 


224 


231 


243 


254 


264 


274 


42 


170 


190 


202 


212 


222 


231 


240 


347 


254 


268 


2S0 


291 


302 


44 


193 


208 


221 


233. 


244 


254 


263 


271 


280 


294 


307 


320 


331 


46 


211 


228 


242 


255 


266 


277 


287 


297 


306 


321 


336 


350 


362 


48 


230 


248 


264 


277 


290 


302 


313 


323 


332 


350 


366 


380 


394 


50 


250] 


269 


286 


301 


315 


328 


339 


350 


360 


380 


397 


413 


427 


52 


270 


291 


309 


326 


340 


354 


367 


378 


390 


410 


429 


446 


463 


54 


291 


314 


333 


351 


367 


382 


396 


408 


420 


443 


463 


481 


500 


60 


360 


388 


412 


433 


453 


472 


488 


504 


519 


547 


571 


594 


616 


66 


435 


468 


'498 


525 


548 


571 


591 


610 


62S 


661 


690 


718 


744 


72 


518 


558 


593 


626 


653 


679 I 


704 


726 


748 


787 


S22 


856 


886 


78 


608 


655 


696 


734 


766 


784 1 


825 


852 


877 


924 


964 


1003 1 


L039 


84 


705 


759 


807 851 


SSS; 


924 


957| 989|: 


L015 ] 


[071 


1116 


1166 


206 


90 


810 


872 


927 975 


L020! 


1062 


1100 1134!' 


L16S ] 


.229 


L284 


1336 ] 


385 


96 


921 


991 


1053,1110^ 


L160| 


1206 | 


L249.1291L 


L327 J 


.4001 


L460 


L505 J 


i575 







Nominal Horsepower < 


)F CC 


NDE47SING 


Engines. 




23 


Stroke of Cylinder Piston S in feet. 


» 


6' 


7' 
H 


W 
H 


1 9' 
H 


10' | 11' 


12' 


: 13^ 14' 


| 15' 16' : 18' 


| 20> 


in. 


~W 


H 


ha~ 


H R 

217i 221 


1 H H 


H~ 


30 


163 


172 


180 


187 


194 


200 


206 


! 211 


1 226 236 


244 


32 


186 


196 


204 


213 


220 


227 


252 


241 


246j 252 


; 25£ 


> 26£ 


278 


34 


210 


221 


231 


240 


249 


257 


264 


272 


276 28c 


291 


30? 


313 


36 


235 


248 


259 


269 


273 


28-8 


29€ 


306 


312 318 


326 


. 33S 


351 


38 


262 


276 


289 


299 


311 


321 


330 


341 


348! 356 


SQl 


378 


391 


40 


290 


306 


320 


333 


344 


355 


366 


377 


3S5 1 395 


! 402 


419 


434 


42 


320 


336 


352 


365 


380 


392 


404 


416 


425 43c 


444 


- 462 


478 


44 


352 


371 


387 


402 


417 


430 


453 


461 


466 477 


494 


507 


525 


46 


384 


405 


423 


440 


460 


470 


484 


497 


510 522 


533 


554 


587 


48 


418 


441 


461 


479 


496 


512 


527 


541 


555 569 


580 


603 


625 


50 


554 


478 


500 


520 


538 


555 


572 


588 


602 ! 617 


j 630 


655 


677 


52 


491 


518 


541 


562 


582 


601 


619 


635 


651 


; 667 


681 


708 


733 


54 


529 


558 


583 


606 


628 


648 


667 


685 


700 


719 


734 


764 


790 


56 


570 


600 


637 


652 


675 


697 


718 


737 


755 


775 


790 


811 


850 


58 


611 


644 


673 


700 


724 


64S 


770 


791 


810 


830 


847 


880 


912 


60 


654 


689 


720 


749 


775 


800 


824 


846 


867 


888 


! 907 


943 


977 


62 


698 


736 


769 


800 


828 


855 


879 


903 


925 


! 948 


' 968 


1007 


1043 


64 


744 


784 


819 


852 


882 


911 


938 


963 


987 


1010 1024 


1073 


1101 


66 


791 


834 


871 


906 


938 


968 


997 


1024 


1049 


1074 1099 


1141 


1182 


68 


840 


885 


925 


960 


996 


1023 


1059 


1087 


1114 


1140 1165 


1211 


1254 


70 


890 


938 


980 


10191055 


1089 


1122 


1152 


1171 


1208 1234 


1233 


1329 


72 


942 


994 


1037 


1078|1116 


1153 


1187 


1218 


1249 


1278 1306 


1358 


1406 


74 


995 


1048 


1095 


1139,1179 


1218 


1254 


1287 


1319 


1350 1380 


1434 


1485 


76 


1050 


1105 


1155 


1201 1244 


1284 


1322 


1358 


1392 


1424 


1455 


1512 


1567 


78 


1105 


1165 


1219 


1265 1310 


1353 


1393 


1430 


1466 


1500 


1533 


1594 


1649 


80 


1162 


1225 


1280 


133111378 


1423 


1465 


1504 


1542 


1578 


1612 


1676 


1737 


84 


1282 


1350 


1411 


1467 1520 


1569 


1615 


1658 


1700 


1741 


1778 


1848 


1914 


88 


1407 


1423 


1549 


1610 166S 


1722 


1773 


1820 


1866 


1909 


1951 


2029 


2100 


92 


153S 


1619 


1693 


1761 


1823 


1882 


193S 


1990 


2039 


20S6 


2133 


225S 


2297 


96 


1674 


1763 


1843 


1917 


1985 2049 


2010 


2166 


2221 


2272 


2322 


2414 


2474 


100 


1817 


1913 


2000 


2080 


21542224 


2290 


2351 


2400 


2466 


2520 


2620 


2714 


104 


L964 


1969 


2163 


2250 


2349i2405 


2477 


2542 


2608 


2666 


2725 


2833 


2935 


108 


2119 


2231 


2333 


2426 


2512 2594 


2671 


274212806 


2S73 


2939 


3056 


3165 


112 


2279 


2399 


2509 


2609 


2702 2790 


2871 


2949 3023 


3092 


3161 


3286 


3404 


116 


2445 


2574 


2691 


2799 


2S98:2992 


3081 


3163,3243 


3315 


3391 


3525 


3651 


120 


2616 


2754 


2880 


2995 


33123202 


3297 


33853471 


3550 


3628 


3772 


3908 


124 


2793 


2941 


3075 


319S 


310113419 


3521 


36143706 


3790 


3S74 


4028 


4172 


128 


2977 


3133 


3277 


340S 


35293643 


3752 


3852 3949' 


4038 


4128 


4292 


4446 


132 


3166 


3333 


3485 


3624 


3753,3875 


3990 


40964190 


4295 


4390 


4565 


4728 


336 


3360 


3538 


3699 


3847 


3984'4113 


4235 


434S 4457 


4557 


4654 


(846 


5019 


140 


3561 


3749 


3920 


4077 


4222J4359 


4488 


46084716 


4832 


4939 


5135 


5319 


144 


3767 


3966 


4147 


4313 


44664611 


4748 


4S75I4997 


5111 


5225 


5432 5681 


148 


3980 


4190 


4381 


1556471814871 


5016 


5179 5279; 


5399 


5519 


57395944 


152 


4198 


4402 


4621 


4805J4976 5138 


5291 


5431 5568 


5696 


5821 


605316270 


158 


4421 


4855 


4S67 


5062:5242 


5412 


5573 


57215865 


6000 


6132 


6376J6604 


162 


4768 


5020 


5249 


545SJ5653 


5S36 


5010 


6170 6324 


6469 


6613 


687617122 


1C8 

174! 


5128 


5399 5645! 


58706079 


3277 | 


6463 


6635 5S02 6958 


7112 


7094 7659 


5494 


5791 6055' 


6297 6521 


5733 


6933 


7117 7296 7464 


7629 


7932 8216 


180 J 


5887| 


6198|6480J 


6539 6979 J 


7205]! 


7419 


7617,7809 7989,8164 


8488,8793 

- 



262 * Horse Power. 



HOE SB POWEE IN MACHINES Y. 

Horse power in Machinery is assumed to be about the effect a horse ig 
able to produce, and has been estimated and establised by Mr. Watt to 
be 33,000 lbs. raised one foot per minute for one horse, which will be the 
same as 550 lbs. raised one foot per second. This applied to steam en- 
gines, a difficulty has been encountered, namely, to fore say the velocity 
of the steam piston, for which Mr. Watt assumed a certain speed for each 
length of stroke, varying nearly as the cuberoot of the length of the 
stroke. He also adopted a standard steam pressure of 7 lbs. per 
square inch, established a simple rule for the nominal horse power of 
engines which is " The square of the diameter of the cylinder in inches multi- 
plied by the cuberoot of the stroke in feet, and divided by the constant number 47 
is the nominal horse power. This rule agreed very near to the actual per- 
formance of engines in those days, but as the improvements advanced 
we found that the steam piston can move with a greater velocity and the 
steam pressure gradually increased, that our days engines greatly exceeds 
the above rule. 

The English Admiralty has adopted Mr. Watt's rule, with a slight 
modification in the assumption of the speed of the steam piston. Area of 
cylinder piston in square inches multiplied by 7 lbs. steam pressure, multiplied 
into the speed of piston in feet per minute divided by 33000. The length of 
stroke and relative speed of piston, and the number of revolutions per 
minute is assumed for 

3 feet stroke 30 revolutions and 180 feet per minute. 
9 " 13^ " 247 " __ " 

in which the speed of the piston will be very near 120 ■*£/$ feet per minute. 
For long strokes the Admiralty's is less, and Mr. Watt's a little more. 

Another expression well known as Indicated horse power, which should 
be understood to be "the gross horse power imparted by the steam on the 
cylinder piston, but in some cases the friction and working pumps about 
10 to 25 per cent, are deducted and still called Indicated horse power. This 
makes a great confusion particularly in datas given of steamship per- 
formance. Numerous other estimates have been made of horse power, 
but most of them founded on that by Mr. Watt. Herein we shall assume 
two kinds, first, Nominal horse power, to express the size, weight and com- 
mercial value of engines, and to be the power spoken of when not ex- 
pressly stated, the second, Actual horse/poiver, which is to express the actual 
power an engine is able to deliver after friction and working pumps are 
deducted. 

Comparing propeller and paddle wheel engines we now find that the 
velocity of the piston is frequently greater for short stroke, still I will 
maintain Mr Watt's rule, because it gives an accurate estimate of the 
real worth of an engine ; shall only alter the coefficient so as to suit our 
days practice. 

NOMINAL HOESE POWEE. 

Assume a standard steam pressure of 30 lbs. per square inch, expanded 
two-thirds the velocity of the" steam piston to be 2C0 \}/S feet and revo- 
. , 100 per minute, we will arrive to a formulae of nominal 

lutions7i = — rsr y - 

V horsepower. H~ _ v ___, for condensing engines, 

which will agree very near with the actual performance of our present 
condensing engines. The preceding tables are calculated from this formulae. 
For high pressure engines I will assume the steam pressure to 80 lbs. 
per square inch, expanded £ which will give the nominal horse power, 

„ £ 2 ^S~ , . . 

E = — - — , nigh pressure engines. 
4 
The horse power in the accompanying table divided by 0-4 gives the 
nominal jpower o high pressure engines. The diameters D are contained 
in the first column in inches, and the stroke S in feet and inches on the 
top line. 



Horse Power. 263 



ACTUAL HORSE POWER 

One actual horse power is 33000 lbs. raised one foot in one minute. This 
applied to steam engines will he the mean steam pressure on cylinder 
piston in pounds, multiplied by the velocity of piston in feet per minute, 
divided by 33,000, is the horse power imparted by the steam. From this 
Ave shall" deduct 25 per cent, in condensing- engines, and 13-1 per cent, 
in high pressure engines, for working friction and pumps, the balance 
to be termed the actual horse power. 

Example 1. Fig. and formulae 318. Area of steam cylinder JL=1809 square 
inches, stroke of piston S=4 feet, indicated pressure of steam 30 lbs. to 
which add the atmospheric pressure 15 lbs. or P=45 lbs. expanded £, the 
mean pressure will be F=31-459 lbs. (see Expansion Table I.), vacuum 
v = 12 lbs. the engine making n=45 revolutions or double stroke per 
minute. Required the actual horse power, H^l JF=31-459-r-12— 14*7= 
28-759 lbs. 1809Y4X28-759X45 

H= ^-^ — ^ - =425-6 horses. 

22000 

In this example the actual horse power is 11*6 per cent, more than the 
nominal power from the table. 

Example 2. Fig. 318. A high pressure engine of cylinder piston .4=314 
square inches, stroke S=3 feet, steam pressure 80 lbs, per square inch, to 
which add 15 lbs. P=95 lbs. expanded &, the engine making n=56 revo- 
lutions per minute. Required the actual horse power? From the ex- 
pansion table we have the mean pressure F= 80-412 lbs., from which sub- 
tract the atmospheric pressure 14-7 lbs. JF=65-712 lbs. 

Tr 314X3Y65-12X56 

S= — ^-^ £ — = 180-8 horses. 

19000 

Annular Expansion Double Cylinder, Fig. 319. 

These kind of engines are now sometimes made in Europe with a view 
to economise fuel, and to extend the expansion of steam. The outer 
cylinder A, A, is annular, similar to that made by Mouslay, but in this 
case it is employed only for expansion, the inner cylinder a is used for high 
pressure only.. It is so arranged by steam valves' that the high steam is 
acting the whole stroke on the small piston a, after which it is conducted 
to the annular cylinder where it acts expansively on the large piston A, A. 
The two pistons being connected by rods to one common crosshead as 
shown by Fig. 319. This arrangement has been successfully carried out 
by Mr. Jagerfelt in Nykoping, Sweden. The inner cylinder can be con- 
sidered an ordinary high pressure engine where the utilized steam is set 
free into the air at each stroke ; but in this case the exhaust steam ac- 
complishes a second engagement in the annular cylinder, which according 
to the grade of expansion may greatly exceed the original effect im- 
parted in the small cylinder during the first engagement. 

Example 3. Fig. 319. Area of the high pressure cylinder piston 
a =254-4 square inches, the annular expansive piston .4=763-2 square 
inches, stroke of pistons 5=3 feet, the high steam pressure P=60 lbs. 
vacuum i*=12 lbs., making n=65 revolutions per minute. Required the 
actual horse power of the engine H='J The grade of expansion will be 

763*2 

1— = §, for which the mean pressure on the annular piston will be 

254-4 

/=32-62 lbs. See Expansion Table II. The effective pressure on the two 
pistons will be F=763-2 (32-624-12— 14-7) +254'4 (60—32-62) = 29800 lbs. 
H= 29800X3X65 =264horseg _ 
22000 
Example 4. Now we will reject the annular expansion cylinder, and 
take the effect of the steam without expansion, when the effectual pres- 
sure will be 60—14-7=45-3 lbs. and the actual power, 

„ 254-4X3X45-3X65 ._ _ 

H = ^ -^ — . = 118 horses. 

19000 



284 Horse Power. 



If we sonsider the last result as unit we shall have 264—118=146 horses 
or nearly 124 per cent, gained by the expansion, omiting the loss of steam 
in the steam passages. 

In the first case about 11 per cent, was gained by vacuum, but that ad- 
vantage is rather in favour of the utility of expansion, because the high 
steam cannot so well be introduced into the condenser. 

The economy will be in the same proportion when the same grade of 
expansion is used in one cylinder. 

I do not mean to maintain that this high per eentage of economy is al- 
ways fully realized in practice, as I am well aware of cases where expan- 
sion is of little use, caused by misconception and carelessness in its em- 
ployment. There are many circumstances about an engine which are in 
favour of expansion, for instance, the steam passages between the main 
valve and cylinder, and the clearance between the piston and cylinder 
heads, contains a great deal of steam which is a total loss, but when ex- 
pansion is used, that steam expands into the cylinder, and is consequently 
utilized. The expanded exhaust require a smaller air pump than would 
be necessary for high steam introduced in the condenser. 

Half Trunk Expansion Engines. Fig. 320. 

This kind of engines has been introduced by Mr. Carlsund, and are ex- 
tensively used in Sweden, they are well suited for Gunboats where the 
machinery is required to be below the water line. The high steam is em- 
ployed throughout the stroke in the annular space around the trunk, 
after which it is conducted to act expansively on the large piston A 
Fig. 320. 

Example 5. Fig. 320. Area of the annular piston #=562 square inches, 
and .4=2248 square inches, stroke of piston <S=4 feet, steam pressure 
P=90 lbs., making ?i=68 revolutions per minute. Required the actual 
horse power] 

Grade of expansion = 1— - — = %, 

From the Expansion Table II. wehave/=41-58 lbs. mean pressure on A, 
The effectual pressure will be F=2248 (41-58— 14-7) +562 (90^41-68) = 
87639 lbs., high pressure 87639X4X68 

H = ^ ^ =627-3 horses. 

38000 

Double Cylinder Expansion Engines, Fig. 321. 

This kind of engines are now made in England and are said to be very 
economical. The small cylinder is used for high pressure, from which 
the steam is conveyed to expand in the large cylinder. In the figure it 
is arranged so that the pistons follow one another in one direction, when 
the steam must be conveyed from the top of the small cylinder to the 
bottom of the large one, and vice-versa ; but it is sometimes arranged so 
that the pistons move in opposite direction, when the steam is conveyed 
direct at the same ends from the small cylinder to the large one, which 
has the advantage of making the steam passages shorter, but is more 
complicated in concentrating the motion. 

Example 6. 

High pressure cylinder, { « = ^square inches. 

Expansion cylinder, { j = fffigg* 1 * inche3 - 
Steam pressure in the small cylinder P=40 lbs., vacuum v=12 lbs., 
making n=21 revolutions per minute. Required the actual horse 
power, H=t 98 2 X 5 

Grade of expansion =1 — — = %. 

X 3848X10 

From the Expansion Table II. we have/=11.879 lbs., mean pressure on A. 
itf=3848xl0 (11.879+12-14:7)4-962X5 (40-11.879) =366767 lbs. of mo- 
mentum. 366767X21 

H= -— = 350 horses. 

22000 



Horse Power of Engines. 



265 



A 



318. One double acting Cylinder* 



Actual \ •£*■ — 
horse •( 



ASWn 



' cond. engs. 



22000 
power. I u=z A SWn . ^S 11 P r - en 



19000 S mes - 

W=F-{-v — 14-7 for cond. engines. 
W=F — 14- 7 for nigh pressure engines . 



319. Annular expansion double Cylinder. 

^=^[2-3 (log.A—log.a)+l]. 
A 

FA— Pa V=A(f+v— 14-7; 

f= A-a ' +<?-/)• 

VSn 




i[h= 



Actual \ **= 22000 

VSn 



, cond. engines. 



power. I 2T=— £, high pr. engs. 



' 19000 



-^ I* 




320. Halftrunk expansion Cy finder. 
F=?£ [2-3(log.A-log.a)+l]. 



f= 



Fa— Pa V=A(f+v—14-1) 

-J^- +a(P-f). 

, „ VSn , 

Actual ( ^^ 44000* 00Ild - en S mes> . 
horse < v 

power. ( H^^^, Ugh pr. engs. 



38000 




321. Double Cylinder expansion. 

F=^-<L2-Klog.AS-log.as)+l] 

, FA— Pa w=AS(f+v— 14-7) 
/= -.- ' +as(P-f). 



A — a 



w n , cond. engines. 



Actual ^"22000 
horse < 

power. I J g- == ZJL, Hgh pr. engines. 
v 19000 ° r 8 



266 Slide Talves. 



SLIDE VALUES. 

The slide valve motion is one of the most important features in causing a 
steam engine to work well, and to employ the effect of steam economically. 
The author of this hook heing well acquainted with disarrangements on this 
point, has here endeavoured to give a good working-drawing of th e proper pro- 
portions and arrangements of slide-valve motion. (See Plate VllI) 

Main Valve* 

It will he best to assume a certain size cylinder, and at the same time give the 
proportions for any size. 

D = 34 inches, diameter of the cylinder. 

S = 18 inches stroke of piston.* 

n = 56 double strokes per minute. 
We have the area of the steamports m, from Formula 26, page 238. 

34».X0-785X18X56 0rt . , , 

a _ 3Q 6QQ = so s^are inches, nearly. 

D+S 34+18 n . . 
* l = -26 =-i-=2mche 8 , 

the width of the steamport ; if the quotient gives a fraction take the nearest 
quarter or eighth. 

— = — = 15 inches, breadth of steamport. 
m S 

r = ±m about = 1 inch, the exhaust port o = 2m — £r = 3f inches, and 
f = o -+• 2r = 5£ inches, h = f — £r = 5£ inches, k = l£m = 3 inches, and 
i = h-{- 2k = 11£ inches, e = m = 2 inches. 

* The stroke and diameter is here rather out of proportion, but we will maintain 
them in the calculations as they suit the drawing, which is purposely made to 
show the slide valves on a large scale. The rules will however suit any propor- 
tions of diameter and stroke. 

To Find the Stroke of the Eccentric* 

s = stroke of the eccentric in inches. 

s = t — / — £r = 5£ inches. 
The lap L = \(i—f— 2m) = I inches. 

The lead of the valve, or opening of the steamport when the crank pin stands 
on the centre should be about 

l = !^» = *f = i inches, nearly. 

Having finished the main valve and ascertained the stroke of the eccentric, 
it is now required to find the position of the centre o, (Plate VI.,) of the eccentric, 
to the crank-pin. Suppose the crank pin of the engine stands at a on the centre 
nearest to the cylinder, and the eccentric rods are attached direct to the valve 
rods; draw the line dd, at right-angle to the centre-line aa" of the engine, 
then 

the angle, Em.W= 2 ^Ml = HSLB) = -409, or TT= 24° 10\ 

See Plates VIII. and IX. 

To Find tlie position of the Crank«Pin at the moment the 
Main Valve opens* 



SI 18X0*25 ... . . 

V " Tc^TF = 5-5X0-9123 - 09 mcbeS ' nearl ^» 



from the centre line. 



Slide Valves. 



Plate T!ffl 




Eccentrics, 



Phi/rlX. 



Fig.l 



Fig. 2. 

F 




Slide Valves. 



267 



To Find the position of the Crank at the moment the 
Exhaust opens. 

x = j(ian.W— i(/— &)1 = ^(o-409~^-(5-5 — 5-25)\ =3-27 inches 

from the centre line. 

To Find the position of the Crank Fin when the Main 
Valve cuts off the Steam* 



v 2SL_ 2X18X1 

-* - T ~ 6-5 = 



: 5-727 inches. 



To Find at what part of the Stroke the Main Valve Cuts 
off the Steam, 
4Z a /oyi\ a 

Will cut off at = 1 =1 — 1—1 = °' 899 of the stroke. 

s* \ 5-5 / 

The greater the lap is, the sooner will the main-valve cut off, hut if the lap is 
increased the stroke of the eccentric must also he equally increased. It does 
not work well to cut off much by the main-valve, especially when the engine 
works fast; for very slow motion it ma/ answer to cut off at $ the stroke. 

It will he noticed that the centre of the eccentric is always ahead of the crank 
pin with an angle 90°+tv. Hence when the engine is to be reversed, the centre 
b must have the same position on the opposite side of the centre-line, or the 
eccentric must be moved forwards an angle of 90° — 2w. 
Cut-off Valve. 

The width of the cut off ports should he about d = fyn = 1£ inch, and their 

breadth — = lT = 12 inches, when two ports are used. 
2d *XJ-+ 

Proportions of the Valve. 

a — b = c — d, a+d = b+c, and a = 2d, and the stroke of the cut-off valve 
eccentric 5 = 2b, we shall have a = 2\, b = 2£, c = 1£, c = H, and 
* = 4$ inches. 

Let us assume the steam to be cut off at \ = I of the stroke S, the position of 
the crank-pin a' will then be sin.u =21 = 0*666, oru = 70° 30 7 ; at the 6ame 
time the position of the centre cf of the cut off eccentric will be 

d+c li+li 

and F= u — z = 70° 30' — 37° htf = 32° 40', the position of the centre c when 
the crank-pin a is on the centre. This Table will show the positions of the 
centre a and c, at different cut offs. Letters correspond with Figure 1, Plate VI. 



= 0-612, or z = 37° 50', 



Cutoff 
at I. 



22° 10' 
32° 40' 
31° 55' 
42° 35' 

46° 30' 
50° 30' 



sm.t> 

0-377 

0-539 

0-527 

0-675 

0-7193 

0-7933 



stroke of 

eccen.s. 

2b 

26 

c+a 

b+c 

a+6 — c 

a+6 — 



z 


[« 


F. 


37° 50' 


60° 


0-5880 


37° W 


70°30 / 


0-6914 


43° 35' 


75°30 / 


0-7332 


47° 25' 


90° 


08350 


58° 


104O 30' 


0-910 


58° 30 7 


109° 30' 


0-985 



0-250 
0-333 
0-375 
0-500 
0-625 
0-666 



It will now be observed that the effectual pressure F in this Table is less 
than in the Table on page 239, owing to the valve not cutting off the steam 
instantly, but gradually, so that the density of the steam in the cylinder is 
already diminished at the cut off point. The valve will cut off quicker the less 
the angle z is. 

See Figure 2, Plate VIILThe actual pressure will not form a sharp corner at 
e, or follow the line e,e,e, as would be due when cut off at | the stroke, but the 
line //'//will be the true diagram. Including the steam in the ports and 
pteamchest, the density at the end of the stroke will correspond nearly with the 
Table. 



268 



Blo wing-off. Salt TV ate. i, Saturation. 



BLOWING-OFF. SALTWATER. SATURATION. 

Sea water contains obout 0-03 its weight of salt. "When salt water boils fresh 
water evaporates, and the salt remains in the boiler, consequently the propor- 
tion of salt increases as the water evaporates, until it lias reached 0-36 weight 
tp the water; the salt will then commence to saturate in the boiler, and the 
water in solution will hold 0'36 weight of salt to 1 of water. 

To prevent this deposit in the boiler, it Is necessary to keep the salt below 
this proportion, which is overcome by withdrawing (blow off) part of the super- 
salted water, while less salted (feed water,) water is replaced. It is found in 
practice that when the proportions are kept 0*12 of salt to 1 weight of water, 
the deposit will bo very slight. To obtain this it will be necessary to blow off 

0-03 

-— - = 0*25 parts of the feed water, or 

if a brine-pump is used, it should be at least 0*25 of the feed pump. 

W= cubic feet of super-salted water to be blown off per minute. 
D, S, n, and k, as before, we shall have, 
D* Sn 



W= 



30007*,- ' 



Example. 


D — 30 inches, stroke of piston 36 


inches, cut off at half stroke 


8= 18, making 14 revolutions per minute, with a pressure of 30 pounds per 


square inch 


, k = 610. How much water must be blown off per minute ? 




^- S Lfx^o 4 - - 12 * cuMcfeet - 




Heat Wasted by Blowing OAT. 




Letters denote. 




7Z fat" WoTnfff* } - ***** *»* P« «"* * time. 




t = temperature of the feed wjffcer. 




T = " " blowing off. 




J3 = heat wasted, per cent. 




W(T-t) 




10(990 + 2'— J)' 


Example. 


Let the quantity of water blown off be \ of the feed water, we have 


TT= 1, and 


10 = 2, the boiling point of the water will then be T= 215-5°, let 


the feed water taken from the hot-well be t = 100°. Required the quantity of 


heat lost ? 






H - 2(990+215-5 -100) " °-° 52 or 5 ' 2 ^ "»* 


This is a very trinng quantity of heat lost. 


Proportion 




Boiling point 


Water blown off. 


Heat lost. 


Specific 


of Salt. 




T. 


per cent. W. 


per cent. 


gravity. 


0-03 




213-2° 


100 


100 


1^03 


0-06 




214-4 


50 


10-35 


1-06 


0-09 




215-5 


33-3 


5-2 


1-09 


0-12 




216-7 


2u 


3-5 


1-12 


0-15 




217-9 


20 


2-66 


1-15 


0-18 




219- 


16-6 


2-14 


1-13 


0-21 




220-2 


14-3 


1-80 


1-21 


0-24 




221-4 


12-5 


1-56 


1-24 


0-27 




222-5 


11-1 


1-38 


1-27 


0-30 




223-7 


10-0 


1-23 


1-30 


0-33 




224-9 


9-07 


112 


1-33 


0-36 




226 


Water sa 


turates. 


1-36 



Blowing off. Salt Water. Saturation. 



259 



Heat wasted by Incrustation* 

The conducting power of iron for heat, is about 30 times that of saturated 
scales, hence a considerable portion of heat is lost when the scales become thick 
in a boiler. 

t — thickness of the scale in 16th of an inch. 
H = heat wasted, in per cent. 



H = 



32-K»* 



Example. The scale in a boiler is 5 sixteenths of an inch thick. How much 
heat is lost by it ? 

5» 
H = = 0*438 or 44 per cent, nearly, 

which goes out through the chimney. 

This is merely to show that the heat lost by blowing off is but trifling, com- 
pared with the heat lost by saturation of scales, which additionally injures the 
boiler by softening and fracturing the iron, and final explosions. 

When boilers are taken good care of by cleaning and blowing off at short 
intervals, the scales need not exceed 1 sixteenth of an inch. 



To Command the Engineer liow to Manoeuvre the Engine 
in a Steamboat} 



I I 



Go ahead - 

Back 

Stop - 

Slowly 

Full speed 

Go ahead slowly 

Back slowly 

Go ahead, full speed -#.-»..#. J. 

Back fast - - J- J- J- J-j- - 

. J33TJ33 



Hurry 



one stroke. 

two strokes. 

one stroke. 

two short. 

three short. 

one long, two short. 

two long, two short. 

one long, three short. 

two long, three short. 

three short repeated. 



23* 



Screw Propellers. 



SCREW PROPELLERS, 

Plate X., is a drawing of a Screw-propeller with proportions thus far known to 
be the most effective, particularly when the steam-engine is applied direct to 
the propeller shaft ; its pitch is twice the diameter at the periphery, but con- 
tracts towards the centre; at the hub the pitch is lessened by the amount of 
slip assumed. AYhen the propeller is geared from the engine, the pitch is gen- 
erally less in proportion to the diameter. 

p = £ of the pitch at the periphery. 
p"= " " " hub. 

s = the assumed slip in a fraction of p. 
Then p = p"+s. 
By these two pitches p andp", the helixes, ac J) at the periphery and def at 
the hub are constructed as for common screws. 

The actual pitch of the propeller at the centre of effort of the blades o is rep- 
resented by p' at r = 0'725jK from the centre ; or the actual pitch = 4p'. 



letters Denote, 

P= pitch of the propeller). th DeriDherv 
W= angle of the blades _ J at me P eri P ner 7- 



D = diameter, R = radius," extreme. 

L = length parallel with the centre-line. 

m = number of blades. 

b = extreme breadth of the propeller blades over the edge, between the 
corners e, e. 

e = circle arc in the angle v. 

v = the projected angle of the blades. 
a = the projected area of the blades. 
A = the true, inclined surface of the blades. 
O = acting area of the propeller. 
H= horse-power required to drive the propeller n revolutions per minute. 

Formulas for Screw Propellers f 



cot. FT- 




~P~ f 



1. 

o 
^i 

3, 

4, 

5, 

6, 



V 129600 

0-785 1) ' v m 
360 ' 



L\ 7, 



8, 



A = R ^ b+L ^'- " * 9 ' 



2-5D' 



10, 



H -mml L 



78 3 / II 

11 ~ D \J (LScos.W 



W-tQ'll) 



12. 



SC3RSW W&fS&SBJ&MfflEL /'/„/, m 





Screw Propellers. 271 



Examplel. The diameter of a propeller is 10 feet 6 inches, and the angle 
Tr== 58° at the periphery. Required the pitch 2^= in feet? 

P= cot.58°X3'14XlO-5 = 20'6 feet. 

Example 2. The propeller on Plate VII. is of dimensions D = 15 feet, L = 5 
feet, W= 57° 30', the slip is 38 per cent: or S = 0*38. What power is required 
to drive it 40 revolutions per minute, H = ? 



^==^000""( 5 X0'38Xcos.57° SO^O-ll ) = 509 horses, nearly. 



vn\ = i 



Example 3. A Propeller of diameter D = 12 feet, angle W— 64°, and length 
L = 3 feet 6 inches, is to be driven by a steam engine of 450 horses, the slip 
S = 0"28. How many revolutions will it make per minute, n = 1 



450 



= 61 revolutions 



= 78 3 f~2_ 

n ~ 12 K/ (3-5X0-28Xcos-64°+0-ll) 
per minute. 

Explanation of Tables. 

Table I. is for finding the pitch and acting area of propellers ; the column 
marked W contains the angle of the propeller blades, as marked on the 
drawing. 

To Find the Pitch* 

RULE. Multiply the diameter of the propeller by the tabular coefficient in 
the pitch column opposite the given angle, and the product is the pitch of the 

propeller. 

Example. The diameter of a propeller is 12 feet, the angle TT= 60°; diameter 
of the hub 1*5 feet, and the angle w = 19°. 

("pitch at the periphery? 
Required the 1 pitch at the hub ? 

I pitch at the centre of effort ? 

Pitch at the periphery = 12X1*814 = 21-768 feet. 
" " hub = 1-5X10-97 = 16-455 feet. 

Let d and p be diameter and pitch of the hub. 

p = pitch at the centre of pressure. We shall have, 

CP-p) : (P —p) = (D-d): (0-725D — d). 

(P__pV0.725.D--d) 
and 9=*+- ~ j)-. d « > 

9 = 16-455+0* 768 ~ "ffj^X" ~ *9 _ 20>097 feet. 

To Find the Acting Area* 

RULE. Multiply the square of the diameter of the propeller by the tabular 
coefficient in the column O opposite the given angle, and the product is tho 
acting area of the propeller. 

Example. The diameter of a propeller D = 13 feet, 3 inches, and the angle 
W= 60°. Required the acting area? 

O = 13-25*X0*679 = 119-2 square feet. 



272 




Table for 


Propellers. 










TABLE I. 






Table for finding the Pitcli and 


Acting Area of 






Propellers* D = 1 


» 




Angle. 


Pitch. 


Act. Area. 


Angle. 


Pitch. 


Act. Area. 


IV 


P 


O 


W 


P 


o 


5° 


36- 


0-068 


47° 


2-930 


0-573 


6 


30- 


0-082 


48 


2-828 


0-583 


r 


25-65 


0-095 


49 


2-730 


0-582 


8 


22-4 


0-109 


50 


2-635 


0-601 


9 


19-85 


0-123 


51 


2-545 


0-610 


10 


17-82 


0*136 


52 


2-455 


0-618 


11 


16-16 


0-150 


53 


2-370 


0-625 


12 


14-79 


0-163 


54 


2-283 


0-634 


13 


13-60 


0-176 


55 


2-200 


0-642 1 


14 


12-60 


0-190 


56 


2-120 


0-650 


15 


11-04 


0-203 


57 


2-040 


0-657 


16 


10-97 


0-217 


58 


1-963 


0-665 


17 


10-27 


0-229 


59 


1-888 


0-672 


18 


9-67 


0-242 


60 


1-814 


0-679 


19 


9-12 


0-255 


61 


1-740 


0-686 


20 


8-64 


0-268 


62 


1-670 


0-692 


21 


8-19 


0-281 


63 


1-600 


0-699 


22 


7-77 


0-294 


64 


1-530 


0-705 


23 


7*40 


0-306 


65 


1-465 


0-711 


24 


7-06 


0-319 


66 


1-400 


0-716 


25 


6-75 


0-331 


67 


1-333 


0-722 


26 


6-45 


0-344 


68 


1-270 


0-728 


27 


6-17 


0-356 


69 


1-205 


0-731 


28 


5-91 


0-368 


70 


1-142 


0-736 


29 


5-67 


0-380 


71 


1-114 


0-741 


30 


5-45 


0-392 


72 


1-021 


0-745 


31 


5-23 


0-404 


73 


0-960 


0-750 


32 


5-03 


0-415 


74 


0-900 


0-754 


33 


4'85 


0-427 


75 


0-842 


0-757 


34 


4-66 


0-439 


76 


0-783 


0-761 


35 


4-50 


0-450 


77 


0-725 


0-764 


36 


4-33 


0-461 


78 


0-668 


0-767 


37 


4-175 


0-472 


79 


0-611 


0-770 


38 


4-025 


0-483 


80 


0-555 


0-772 


39 


3-885 


0-494 


81 


0-498 


0-775 


40 


3-745 


0-504 


82 


0-442 


0-777 


41 


3-620 


0-515 


83 


0-386 


0-779 


42 


S-500 


0-523 


84 


0-331 


0-780 


43 


3-370 


0-535 


85 


0-275 


0-781 


44 


3-260 


0-545 


86 


0-220 


0-782 


45 


3-141 


0555 


87 


0-165 


783 


46 


3-035 


0-564 ■ 88 j 


0-110 


0-784 



Coefficient of Vessels. 



273 





TABLE II. 








Tabic for finding the Exponent and Coefficient of Vessels* 


Full Lines. 




Hollov. 


Lmes. 




Exponent x. 


Coefficient Jc. 


Exponent x. 


Coefficient k. 


1 


o-ooo 




0-68 




1-71 


0-95 


0-024 




0.67 




1-77 


0-90 


0-228 




0-66 




1-84 


0-88 


0-326 




0-65 




1-90 


0-86 


0-432 




0-64 




1-96 


0-84 


0-558 




0-63 




2-00 


0-82 


0-692 




0-62 




1-97 


0-80 


0-836 




0-61 




1-93 


0-79 


0-902 




0-60* 




1-88 


0-78 


0-978 




0-59 




1-82 


0.77 


1-050 




0.58 




1-77 


0-76 


1-12 




0-57 




1-72 


0*75 , 


1-20 




0-56 




3-67 


0-74 


1-28 




0-55 




1-61 


0-73 


1-35 




0-54 




1-55 


0-72 


1-43 




0-53 




1-50 


0-71 


1-51 




0-52 




1-44 


0-70 


1-59 




0-51 




1-38 


0-69 


1-64 




0-50 




1-32 




TABI 


E III. 








TaTble for t 


siding tlie Slip 


and - 


Acting Area* O = !• 


blip. 


Act. Area 


Slip. 


Act. Area. 


Slip. 


Act. Area. 


Slip. 


Act. Area. 


S. 


o 


S. 


o 


s. 


o 


S. 


O 


per cent. 




per cent. 




per cent. 




per cent. 




5 


84-85 


28 


4-150 


51 


0-927 


74 


0-208 


6 


60*35 


29 


3-S20 


52 


0-888 


75 


0-192 


7 


46-35 


30 


3-555 


53 


0-840 


76 


0-177 


8 


39-00 


31 


3-333 


54 


0-784 


77 


0-163 


9 


32-20 


32 


3-090 


55 


•0-737 


78 


0149 


10 


27-00 


33 


2-880 


56 


0-697 


79 


0-137 


11 


22-07 


34 


2-710 


57 


0-655 


80 


0-125 


12 


19-80 


35 


2-535 


58 


0-611 


81 


0-113 


13 


17-32 


36 


2-366 


59 


0-581 


82 


0-103 


14 


15-20 


37 


2-222 


60 


0-546 


83 


0-092 


15 


13-52 


38 


2-085 


61 


0-511 


84 


0-083 


16 


12-00 


39 


1-952 


62 


0-479 


85 


0-074 


17 


10-82 


40 


1-827 


63 


0-455 


86 


0-066 


18 


9-715 


41 


1-727 


64 


0-422 


87 


0-058 


19 


8-820 


42 


1-634 


65 


0-394 


88 


0-050 


20 


8-000 


43 


1-523 


66 


0-369 


89 


0-045 


21 


7-282 


44 


1-430 


67 


0-347 


90 


0-037 


22 


6-700 


45 


1-354 


68 


0-323 


91 


0-031 


23 


6-111 


46 


1-275 


69 


0-300 


92 


0-026 


24 


5-635 


47 


1-195 


70 


0-281 


93 


0-021 


25 


5-200 


48 


1-127 


71 


0-263 


94 


0-017 


2Q 


4-785 


49 


1-068 


72 


0-241 


95 


0-012 


27 


4-444 


50 


1-000 


73 


)-225 







Steamship Performance. 



STEAMSHIP PERFOEilAXCE. 

I-'.' '.'•;■;- 5 at : . ::. 
T = Displacement of the vessel in tons. 
JS= greatest immersed section in square feet. 
gj = area of resistance in square feet. 
/ = length of the vessel in feet, in the loadline. 
b = : tam 

F = resistance of the vessel in pounds, including friction of the im- 
mersed surface. 
k = coefficient, .r=exponent of the vessel. See Table II. preeedin? page. 
H= Actual horse power required to propel the vessel. 
M= nautical miles or knots per hour. 
35 T 
*=£y, - - - % - 1 F=* 2 yp f .... 3 




/ — r- — ^v 3 

>3Cy ' b ' , - 2 H=:r V- " -■- - 4 

\ l—k r' bl 

w r.r .:-:;: h l. The I". S. steam Frigate Nlag 

-X_ K 1=338*9 feet long. 5=55 feet wide, greatest inun 

"" * ' section J£=S55 square feet, di« 9 tons. 

^Required what horse power is necessary to 
her 37=10 knots per hour in smooth water. 

35 \ 5000 

Exponent z = - — = 0-63 nearlv, 

S55\ 325-9 

for which the coefficient in Table II. preceding page is k=2. From 
formulae 2 we have the area of resistance. 

/ 55- 

ft = B66 v / - — c = 104 square feet. 

\ 55-— 2 \ 325' 9- 

TT 104\ 10 3 „ , 

Actual power H = — — — = 1284 horses. 



V 2. A barge of 7=165 feet Ion?. 6=25 feet wide. 
^— ~^~ -"--^-"ofossection J5T=ll2 square feet, dis] r=4l0 tons. 



How many horses are required to draw the barge at a speed of 31=2 
miles per iiour in a canal ) 

38* N'-llO 

Exponent x = - -- = 0*8. Coefficient £=0-836. 
* 113V16G 



Area of resistance S = 112 V / = 65 '" square feet. 



\ 2-5' : — 0-S36X1 



Tractive force F=4X65 , 7\c-=i050 pounds. Sea utiles 8=2*3 statute 
miles. See page 123 for ability of a horse working <?=5 hours continually. 

F= — — = 73 pounds. 
2'3 | 5 
Number of horses 1050:73 = 14-36 or 15 horses are required to draw the 
barge 8*3 statute miles pei 

The mechanical horse power performed will be, 

_ 65 -7 \ 2 3 

H= — — = 6-o horses. 

SI 
If the breadth of the canal is tee times that of the barge the 

resistance will be more, and require more horses. 

For similar proportioned vessels the resistance is a function of £*3/ , 
and the horse power a function of JST 3f z - The displacement of a vessel is ■ 

* 35 for salt water, and 36 for fresh water. 



Steamship Performance 275 



function of the cube of any linear dimension of the same, and the gw 
immersed section J£*a function of the square of any linear dimension of 
the displacement, consequently the V T is a function of any linear di- 
dimension, and (\?>T ) a =V ^T 2 is a function of j£ ; therefore, the 
ance is a function of if^&Fj and the horse power a function of.3: 

thus we arrive at what is known as Mr. Atherton's formula C= _. t .'- 

H 
about which so much controversy has existed in English Journals. 
The above formulas l. 2, 3. and 4 will give the same result for - 
vessels as that of Mr. Atherton's. and it wiU be found that they _-: T . e dif- 
ferent coefficients for different proportions of vessel as seen by the ti : ex- 
amples where the quality of the performance is considered to'be the same. 

Ex.l. C = I^°^ = 223and Ex. % C = ^^ = 63 as coeffl. 
1284 6-5 

These coefficients are very small compared with that of Mr. Atherton's 
on account of the different "estimate of horse power ; it can however in 
neither case be considered a measure of the quality of performance for 
different proportioned vessels ; neither can it be considered a measure of 
commercial value, because the commercial effect produced will be. 

Example I. 10X5OO ° =Z9. Example 2. ?- X4 - = 126 effects, 

r 1284 6-5 

which is quite the reverse of the coefficient. 

The following Table is calculated from the formulas 1, 2, 3. and 4. for dif- 
ferent sizes of vessels of similar proportions as that of Niagara. If those 
formulas are well understood it will be found that they trace a line of 
justice between the Engineer and ship builder, that when the perform- 
ance is known it shows "from where the fame or blame is due. 

River steamers of light draft and flat bottom requires more power for 
the same sharpness of lines, as will be found by the formulas. 
To Approximate the size of Steam Engines. 

Example 3. It is required to build a river steamer of displacenent 
T=1000 tons, to run i/=16 nautical miles per hour. Required the size 
of the cylinder for an ordinary overbeam engine ? From the table of 
steamship performance will be found the required actual power 11=1795 
horses. 

From the table of Nominal horse power select the approximate size of 
cvlinder which may be D=SS inches, diameter of cylinder by S=14 feet 
stroke, which answers to H=1S66 horses nominal. In this case the nomi- 
nal horse power can be considered the same as the actual. 

Li'cmple 4. A propeller steamer is to run 32=10 nautical miles per hour, 
with a displacement T=3400 tons. Required the size of the cylinders ! 

From table of steamship performance if =992 horses, to be divided into 
two cylinders of 496 each. Select from table of Nominal horse power 
D=60 inches diameter of cylinders and 5=2' 9" stroke of piston, which 
answers to H=604, or 504X2=1008 horses of the two cylinders.' After 
these approximations are made, make a careful calculation from the 
original formulas. 

Ea ample 5, Suppose the propeller for the steamer in the preceding ex- 
ample 4 makes n = Q0 revolutions per minute. Required the diameter of 
the propeller shaft ! See Table, page 176 for wrought iron shafts, for 500 
horses and 60 revolutions, the shaft should be 101 inches. Multiply this 
by the cuberoot of 2, or 10 , lXl , 26=12-926, say 13 inches the diameter 
required. 

npie 6. A steamer of T=2500 tons is to run M—9 nautical miles per 
hour with an indicated steam pressure of 20 lbs., or P=35 lbs. per square 
inch, expanded t- Required the consumption of fuel in tons per 24 hours I 

Table of steamship performance H=5S5 horses. 

Table V., page 258 consumption of fuel. 3-44 tons. 

The required consumption will be 5 , S5X3-44=20 , 124 tons per 24 hours, 
steaming. 



276 






Steamship Peri 


-ORMANCE. 








|i 




Nautical Miles or Knots p 


er Hour. 




Displace-, 1 
ment. j 


5 1 


6 | 


J 1 


8 | 


9 


10 | 


11 


12 


T 


H 


H 


H 


H 


H 


H 


H 


H 


100 


11-8 


19-4 


32-4 


48-4 


68-5 


94*5 


126 


163 


,200 


18-8 


32-5 


51-5 


76-9 


110 


150 


200 


260 


300 


24-5 


42-4 


67-5 


100 


142 


196 


262 


340 


400 


29-8 


51-4 


81-7 


122 


172 


238 


317 


412 


500 


34-6 


59-6 


94-3 


141 


200 


276 


368 


478 


600 


39-0 


67-2 


107 


160 


226 


313 


415 


540 


TOO 


43-3 


74-6 


119 


177 


250 


377 


460 


599 


800 


47-3 


81*5 


130 


194 


274 


378 


503 


654 


900 


51-1 


88-1 


140 


210 


296 


409 


545 


708 


1000 


54-9 


94-6 


150 


225 


318 


439 


585 


759 


1100 


58-4 


100 


160 


239 


338 


467 


622 


806 


1200 


62-0 


107 


170 


254 


359 


495 


660 


858 


1300 


65-3 


112 


179 


267 


378 


523 


696 


903 


1400 


68-7 


119 


189 


281 


398 


549 


732 


950 


1500 


71*9 


124 


197 


295 


417 


575 


766 


995 


1600 


75-0 


130 


206 


307 


435 


600 


800 


103S 


1T00 


78-1 


135 


215 


320 


453 


625 


833 


1083 


1800 


81-2 


140 


224 


332 


470 


649 


864 


1123 


1900 


84-2 


145 


231 


345 


488 


673 


897 


1166 


2000 


87-0 


150 


239 


356 


504 


696 


927 


1205 


2100 


90-0 


155 


247 


369 


521 


720 


958 


1247 


2200 


92-7 


160 


255 


380 


537 


741 


988 


1284 


2300 


95-6 


165 


262 


391 


554 


764 


101.7 


1324 


2400 


98-4 


170 


270 


402 


569 


786 


1047 


1360 


2500 


101 


174 


277 


414 


585 


808 


1077 


1400 


2600 


104 


179 


285 


424 


600 


828 


1102 


1435 


2700 


106 


184 


292 


436 


616 


850 


1131 


1473 


2800 


109 


188 


299 


446 


631 


871 


1160 


1508 


2900 


111 


192 


306 


457 


646 


893 


1189 


1545 


3000 


114 


197 


313 


467 


660 


913 


1215 


1582 


3100 


117 


201 


320 


478 


676 


933 


1242 


1614 


3200 


119 


205 


327 


488 


690 


952 


1268 


1648 


3300 


121 


209 


334 


498 


704 


972 


1296 


16S3 


3400 


124 


214 


340 


508 


718 


992 


1320 


1717 


3500 


127 


218 


347 


518 


733 


1010 


1347 


1750 


3600 


129 


222 


354 


528 


746 


1025 


1373 


1783 


3700 


131 


226 


360 


538 


759 


1049 


1398 


1815 


3800 


133 


230 


367 


548 


774 


1070 


1422 


1848 


3900 


135 


234 


373 


558 


787 


1087 


1446 


1880 


4000 


138 


238 


380 


507 


801 


1105 


1473 


1912 


4100 


140 


242 


386 


577 


814 


1122 


1497 


1944 


4200 


142 


246 


392 


586 


827 


1141 


1520 


1975 


4300 


145 


250 


398 


595 


840 


1160 


1545 


2008 


4400 


147 


254 


404 


604 


853 


1179 


156S 


2037 


4500 


150 


258 


410 


613 


866 


1198 


1593 


2070 


4600 


152 


261 


416 


622 


879 


121(5 


1614 


2100 


4800 


156 


270 


i 428 


640 


904 


1248 


1663 


2160 


5000 


160 


277 


440 


658 


929 


1282 


1708 


2220 


5500 


171 


295 


I 469 


700 


990 


1367 


1822 


2365 


6000 


181 


303 


i 497 


742 


1050 


1448 


1930 


2507 







Steamship Performance. 




277 






Nautical Miles or Knots per Hour 


• 




Displace, 
meut. 


13 


14 


15 


16 


IT 


18 


19 


20 


T 


H 


H 


H 


H 


H 


H 


H 


H 


100 


207 


259 


318 


387 


464 


551 


648 


756 


200 


329 


4L2 


506 


615 


737 


875 


1027 


1201 


300 


432 


540 


662 


806 


966 


1146 


1347 


1573 


400 


522 


654 


803 


976 


1170 


1402 


1632 


1907 


500 


607 


759 


932 


1131 


1358 


1611 


1896 


2213 


600 


684 


856 


1036 


1280 


1532 


1820 


2140 


2500 


700 


759 


938 


1166 


1417 


1700 


2016 


2373 


2770 


800 


830 


1038 


1274 


1548 


1857 


2206 


2593 


3026 


900 


898 


1123 


1380 


1675 


2009 


2385 


2803 


3274 


1000 


963 


1206 


1480 


1798 


2157 


2560 


3008 


3514 


1100 


1024 


1284 


1574 


1913 


2295 


2723 


3203 


3736 


1200 


1090 


1360 


1670 


2030 


2435 


2890 


3400 


3967 


1300 


1147 


1432 


1758 


2136 


2564 


3043 


3576 


4178 


1400 


1204 


1508 


1850 


2248 


2697 


3200 


3762 


4394 


1500 


1264 


1580 


1938 


2355 


2825 


3352 


3943 


4605 


1600 


1317 


1648 


2020 


2458 


2948 


3500 


4113 


4S03 


1700 


1374 


1718 


2107 


2561 


3072 


3646 


4286 


5006 


1800 


1422 


1784 


2188 


2660 


3190 


3785 


4448 


5195 


1900 


1479 


1850 


2270 


2760 


3310 


3928 


4615 


5390 


2000 


1527 


1913 


2345 


2854 


3420 


4060 


4770 


5570 


2100 


1582 


1979 


2382 


2948 


3535 


4195 


4935 


5762 


2200 


1628 


2037 


2500 


3038 


3642 


4325 


5084 


5935 


2300 


1680 


2102 


2578 


3134 


" 3755 


4460 


5241 


6120 


2400 


1723 


2160 


2646 


3220 


3860 


4580 


5386 


6290 


2500 


1777 


2222 


2725 


3313 


3970 


4715 


5542 


6470 


2600 


1820 


2280 


2796 


3400 


4075 


4835 


5655 


6637 


2700 


1870 


2338 


2868 


3486 


4180 


4960 


5832 


6813 


2800 


1911 


2395 


2935 


3568 


4280 


5076 


5970 


6970 


2900 


1960 


2452 


3010 


3655 


4385 


5200 


6115 


7142 


3000 


2000 


2508 


3075 


3740 


44S5 


5318 


6255 


7300 


3100 


2048 


2565 


3145 


3822 


4585 


5440 


6394 


7470 


3200 


2092 


2616 


3210 


3905 


4680 


5550 


6525 


7622 


3300 


2134 


2671 


32S0 


3985 


4775 


5670 


6666 


*7781 


3400 


2178 


2725 


3343 


4063 


4870 


5784 


6784 


7936 


3500 


2220 


2779 


3408 


4143 


4965 


5S93 


6936 


S090 


3600 


2264 


2830 


3475 


4222 


5060 


6010 


7061 


8250 


3700 


2303 


2881 


3534 


4300 


5155 


6115 


7184 


8400 


3800 


2348 


2941 


3606 


43S5 


5250 


6238 


7333 


8563 


3900 


2385 


2986 


3660 


4453 


5340 


6336 


7444 


8695 


4000 


2427 


3038 


3725 


4530 


5430 


6444 


7580 


8847 


4100 


2468 


3086 


3785 


4610 


5520 


6550 


7700 


8988 


4200 


2507 


3137 


3850 


4680 


5610 


6655 


7830 


9141 


4300 


2546 


3186 


3910 


4750 


5700 


6761 


7950 


9285 


4400 


2585 


3238 


3970 


4825 


5790 


6865 


8072 


9432 


4500 


2624 


3286 


4025 


4900 


5875 


6970 


8195 


9572 


4600 


2664 


3333 


4087 


4970 


5960 


7070 


8320 


9710 


4800 


2740 


3431 


4202 


5113 


6130 


7275 


8555 


9990 


5000 


2S17 


3525 


4321 


5253 


6300 


7475 


8792 


10250 


5500 


3000 


3755 


4608 


5600 


6715 


7972 


9370 


10953 


6000 


3180 


3981 


i 4880 


5935 


7120 


8446 


9935 


11586 



24 



Tonnage of Vessels. 



Speed of Steantboats* 



Screw Propellers. 



Paddle-Wheels. 



100 1( > 



s = l 



IQOitf 

P(l — S)' 

100M 



Pn' 



2, 



3, 



16 M 



R cos. J"W(1 — 5)* 
16M 



5 = 1 — 



ni?cos.jW 



Example, The pitch of a propeller is P = 30 feet, and makes 40 revolutions 
per minute ; the slip is S = 0-4. Required the speed of the vessel ? 

Jf = ° - * Q (1 — 0-4) = 7-2 miles per hour. 

Explanation of Table III. 

To find the Slip. 
RULE. Divide the acting area of the propeller, or paddle-wheels by the area 
of resistance of the vessel ; find the quotient in the columns of acting area, and 
opposite, in the slip column, is the slip per cent. 



TONNAGE OF VESSELS. 

The United States Custom House measurement of tonnage of vessels is 
expressed by the formula. 

bd 

95 

Letters Denote, 

T ' = tonnage of the vessel. 

b — extreme beam in feet, taken above the main-wales. 

d — depth of the vessel in feet. In double-decked vessels half the beam b is 
taken as the depth. For single decked vessels, the depth is taken from the 
under side of the deck plank to the ceiling of the hold. 

I = length of the vessel in feet, from the fore-part of the steam to the after part of 
the stern-post, measured on the upper deck. 

Example. The dimensions of a vessel are I — 186 feet, = 30, and d = 15 
feet, for a double-decked vessel. Required her tonnage ? 



T = 



30X15 
95 



(l86-f 3o) = 



795*77 tons. 



To Approximate the Tonnage of the Displacement* 

n _ Wis/ / 0-015 very sharp vessels. 
V - ^X \ 0-022 very full vessels. 



Fresh "Water Condensor. 



279 



FRESH WATER CONDENSOR. 



Fig. 1. 




Fig. 2. 



o o o o o O CO 
oooooooo 

OQOOOCOO 
OOOOTOOO 

oooooooo 
oooooooo 
oooooooo 
oooooooo 
.oooooooo 



Fig. 1, is a longitudinal, and Fig. 2, a transverse section of a fresh water con- 
denser with horizontal tubes. 

A, air-pump, a, fresh water. J^ exhaust-pipe. 6, hot well. T f tubes. 
c, injection pipe, d, strainer. 

The tubes are of copper one inch outside diameter, thickness of metal, 
No. 22 or 24 wire guage, weighs b~ ounces per foot. The space occupied by 
the tubes should be about cubical, that is, the sides of the tube-plate should be 
about the lerigth of the tubes. 

Between the injection and the tubes is a horizontal strainer, to spread the cold 
water uniformly over the tubes. Steam inside the tubes. 
Letters Denote. 
A = condensing area of all the tubes in square inches. 
I = length of tubes, height and breadth of tube-plate in inches. 
2T= number of tubes in the condensor. 

D= diameter of steam cylinder. S = stroke of piston, in inches. 
n — number of revolutions of engine per minute. 

T— temperature of exhaust steam ) M „ . . .- _ - ni0 

k =jroluuni coefficient of steam / 8ee steam table > **& 248 « 

A ~ P * 6 W k n (teQ+T) } l = 0-853^, &= 0-5128 7*, A =» 1-61 P. 

Example. A "fresh water condensor is to be constructed for an engine of 
D = G2 in. S= 7G in. making n = 34 revolutions per minute J T=230°, fc=1225. 
Required the condensing area of tubes, A = ? 

A=Z ^J^^P^ (940+230°)=295,709 square inches. 
Required the length of tubes, and sides of tube plate, I = ? 

I = 0.853^295,769 = 57 inches nearly. 
Required the number of tubes in the condensor, 2V= ? 
iV = 0-5128X57 3 = 1666 tubes. 

Number of tubes in the top tow r-r=^r~ = 38* side row =+-?-=* 44 tabes. 
r 1"5 1'5 l'o 

The tubes to be placed zigzag, as shown in Fig. 2. Should the location for 
the condensor not permit the cubical form, we have, 

Length tubes I-y^g, Breadth of tube plate o - ^ Heightft^-^ 

Fresh water produced G = ■ *- , n , gallons per minute, or about 75 per cent 
lbi k of the feed water. 
Temperature in the hot well about 110° to 115°. 

„ cf fresh water 120° to 130°. 

For fresh water condensors the capacity of the air pump should be about 
10 pel cent larger than by the rules on page 236. 



280 



Steam boilers. 



STEAM BOILERS. 



The accompanying proportions are averages of a great number of good 
marine boilers. 

Letters denote. 

D = diameter of the steam-cylinder in inches. 

S = stroke of piston under which steam is fully admitted, in inches. 

n = number of double strokes, or revolutions per minute. 

V) = pounds of water evaporated per pound of coal, per hour. 

k = volum coefficient from the steam table. 

EEEI = fire grate in square feet, for each cylinder, and with natural draft. 

To Find the Area of Fire Grate. 



; X>* Sn 

4:'Q6wk' 



n =- 



4-66 wH 



D* S 



1,2. 



Example 1. A steam engine of D = 54 inches diameter of the cylinder, and 
stroke of piston 96 inches, cut off at |, £=48 inches ; is to make 22 revolutions 
per minute. Anthracite coal to be used, that evaporates w = 7 pounds of 
water per pound of coal, and to carry 27 pounds of steam per square inch, 
k ■*=■ 649. Required the area of fire grate EE3 = ? in square feet. 



, = 54*X48X22 
4-66X7X649 



= 145*34 square feet. 



'Example 2. A steamboiler of F=i = 128 square feet, is to be used for an 
engine of D = 36 inches diameter, and 64 inches stroke,— cut off the steam at 
I then S = 42*66 inches. Steam pressure to be kept at 25 pounds per square 
inch k = 679. w = 6*5. Required for how many revolutions per minutes can 
the steam be kept at 25 pounds 2 



4-66X6-5X679X128 
36 2 X42-66 



= 47*6 revolutions. 



Horse Power of the Fire Grate, 

JET = horse power of the fire grate. 

p = pressure in the boiler in pounds per square inch, excluding tho 

atmosphere. 
jp» vacuum in the condensor in pounds per square inch. 

Hx 
13=3 kw{P+0.8p)' x 

f i the stroke, x = 27700. saves 55 
x = 31400. „ 49 
x = 38400. 
x = 45500. 
x = 49100. 
Steam admitted throughout the stroke x = 61700, 



jr „ EEU-»(i»+0.8p) . 3 4# 



Cut off the 
steam 



Liie J | : - 

at 1 I » 

3 5» 

U n 



per cent 

26 of fueL 
20 
per cent. 



Example 3. Steamboilers are to be constructed for an engine of 650 horses, 
the steam to be cut off at h the stroke, and P = 36 pounds per square inch, 
Jc = 544, to = 7*5 pounds of water evaporated per pound of coal. Required the 
fire grate in the boilers £=l = 1 in square feet. 

, 650X38400 

13 = 554x7.5(36 + 0.8X11) - 136 sc l uare feet 



Steam boilers. 281 



Example. 4. Required the horse-power of a fire grate KEEEI = 112 square feet, 
to carry 18 pounds steam, and cut off at \ the stroke 1 k — 810, w = 7 pounds. 

rr 112X18X810X7 „„ , 

JI --= — =2ol'2 horses. 

45500 

Consumption of Coah 
C= coal consumed in pounds per hour. 

^ 3 D* Sn „ UHx - fl 



wi & w (P+0'8^) 

Example 5 A steam engine of 2) = 42 inches diameter, and 48 inches stroke, 
cut off the steam at | # = 16 inches, is to make n = 65 revolutions per minute 
with a pressure of 34 pounds per square inch, k = 564, and w = 6 pounds. Re- 
quired the consumption of coal in pounds per hour C = ? 

n 3X42*X16X65 ™- , , 

(7= ., . . = 162o pounds per hour. 

6Xt>64 

Example 6. A pair of steam engines of H= 260 horses are to be worked with 
P = 28 pounds per square inch, cut off at f the stroke, k = 635, the coal to 
evaporate w = 6-5 pounds of water per pound of coal. Required the consump- 
tion of coal in pounds per hour C — ? 

<7- 1^260X31400 = m unds w 

630X6.5(28+0.8X10) 

It will be observed in the formulas 4 and 6, that the higher steam used, the 
less fuel and fire grate is required for the same power,— the proportion of fuel 
will be nearly as the square root of the steam pressure ; and still more fuel is 
saved by cutting off the steam at an early part of the stroke. 

Fire Surface* 

1. In the common single returning flue boilers, the whole amount of fire 
surface from the grate to the water line should be about 25 B — a . 

2. In flue and returning tubular boilers the whole amount of fire surface 



3. Boilers with vertical flues, (2 inches diameter,) fire outside and water in- 
side, fire surface 35 B a . 

Area of Flues* (Calorimeter.) 

In the common single returning flue boilers, the area of the first row should 
be about 0.18 *=% . 
Returning row, (flue or tube) 0.13 E3 
Area of chimneys 0.16 F=j . 

Height of Chimney. 

C=2H3V / 2 : H, &= ^ a 2. 

Example. Required the height of a chimney for the consumption of 15 
lbs. of coal per square foot of grate. 

height, h = *5! —2=54 feet, 

4X1- 



282 




Fuel and Iron Tubes. 










Properties of Fuel, 












u 

III 


o 




u . 


o . 


Kind of Fuel. 




0> 




■ssl 

.2 o 

£2 


•2 


42 S3 

-°-2 


Bituminous coal, 


_ 




7 to 9 | 80 


265 


50 


44 


Anthracite coal, - 






8 to 10 99 


282 


54 


40 


Coke, - - - - . - - - 


8 to 10 


86 


245 


31 


72 


Coke, nat. Virginia, 


- - - . 




8 to 9 


80 


260 


48 


48 


Coke Cumberland, 


- - - . 




8 to 10 


80 


250 


32 


70 


Charcoal, - 






5 to 6 


98 


265 


24 


104 


Dry wood, ------- 


4 to 5 


44 


147 


20 


100 


Wood with 20 per ct. 


water, 




4 


34 


115 


25 


100 


Turf dry, - - - 






6 


51 


165 


28 


80 


Turf 20 per ct. water, 


- 




5 


40 


132 


30 


75 


Illuminating sras - 






13*8 




194 


0-037 


29800 


Oil, Wax, Tallow, 


- - 




14 


77 


200 


59 


37 


Alcohol (from market.) - - - 




9-56 


58 


154 


52 


42 


Chemically one pound of carbon burnt to carbonic acid requires the 


oxygen of 153 cubic feet of atmospheric air. 




Morris Tasker & C 


o., Pascal Ironworks, Philada. Jan. 


1861, 


Lap-welded American 


Charcoal Iron Boiler Flues 










Extra Price, 










Each safe Screw on 




Out3ide 


Thickness of 


Heating sur- 


Weight 


Price 


end col- 
lar or 


one end 
collar 


cross 
flues 


diameter. 


iron. 


face per ft 


per ft 

pounds. 


P 

$ 


2r ft- Phoulder 


other. 


each. 


inches. 


Wg. inches. 


Sq. ft 


. c. 


$. c. 


$. c. 


$. c. 


. 1-25 


15 0-072 


0-3273 


1*12 


18 


16 


82 


1*45 


S 1*5 


14 0-083 


0-3926 


1-40 


22 


16 


82 


1-75 


S 1-75 


13 0-095 


0-4589 


1-60 


24 


18 


86 




•§ 2 


13 0-095 


0-5236 


1-95 


28 


18 


86 


2-50 


§ 2-25 


13 0-095 


0.5890 


216 


31 


20 


90 


2-75 


ft 2-5 


12 0-109 


0.6545 


2-60 


33 


23 


96 


3-00 


=g 2-75 

s 3 


12 0-109 


0.7200 


2-98 


36 


26 


1-02 




12 0-109 


0-7853 


3-16 


41 


29 


1*08 


4-00 


3 3-25 


11 0-120 


0-8508 


3-78 


46 


32 






| 3 ' 5 


11 0-120 


0-9163 


4-21 


60 


35 






to 3*75 


11 0-120 


0-9817 


4-90 


70 


38 


F4 


£ 


© 4* 


10 0-134 


1-0472 


5-25 


75 


40 


*& 


•a 


o 4-5 


10 0-134 


1-179 


5-54 


1-00 


43 


O 


o 


S 5 


9 0-148 


1-3680 


6-48 


115 


45 


O 


o 


1 6 

m 7 


8 0-165 


1-5708 


10-0 


1-65 


50 




T3 


8 0-165 


1-8326 


12 


2-00 


60 




c3 


3 8 


8 0-165 


2-0944 


14 


2-75 


75 


"S 9 


7 0-180 


2.3562 


17 


3-50 








« 10- 


6 0-203 


2-5347 


20 


4-25 








" Sa/e Ends" of thic 


ker iron v 


relded to one or both e 


nds of f 


lues to 


order, gives increasec 


1 strength 


to the flue for calkir 


g in th( 


i tube- 


plate. The addition* 


il thicknes 


3s of the safe end is pi 


aced in 


or out- 


side of the Flue, as i 


nay be spe 


cified in the order. 







Combustion and Effect of Fuel, 



283 



Combustion is the rapid chemical combination of substances with ( 
oxygen. Carbon C and hydrogen H, are the substances most generally 
employed for generating heat. Carbon is fully consumed when combined 
with oxygen 0, to form carbonic acid gas CO2, and partly consumed 
when in the form of carbonic oxide gas CO or smoke. /i=units of heat 
generated of one pound of fuel. The heat necessary to raise one pound 
of water one degree Fah. is one unit of heat. 7#=pounds of water at 
212° evaporated per pound of fuel. ,4=volume in cubic feet and a=weight 
in pounds of atmospheric air required for the perfect combustion of one 
pound of fuel. C, 0, and H are in the four first formulas, fractions in 
one pound of the compound fuel. 



Imperfect Combustion. 

{ co)^ c - 210 



(COJ 



12 

330- 44 Q 
12 ' 



- 6 



Perfect Combustion. 
J 4==149[CH-3(fl-|)],- - 1 

«=12[C+3(IZ-?)1 - - 2 
7i=14500[C4-4-28fl— §)], 3 

»=-^6=15[CH-4-28(fl-|)] 

When oxygen is supplied to carbon in a proportion between CO and 
C0 2 , both the gases will be formed separately in the proportion of the 
formulas 5 and 6, when the heat generated will be as formulas 7 and 8, in 
which C, 0, CO and COi are expressed in pounds, for instance : 0=20 lbs. 
of oxygen united with G=:12 lbs. of carbon will form 



7i=3960(tfaj+1650(C<9), - 
^=8002-50—6820(7, - - 



(00)=- 



56X1 2— -21X20 _ 
"12 ~ 

33X20—44X12 
= 12 = 



:21 lbs. carbonic oxide or smoke, 



=11 lbs. carbonic acid, and will 



(c6*y 

generate 7i==8002'5X20— 6820X12=78210 units of heat. 

One unit of heat=772 foot pounds, if generated per second will be H= 
1.4 horses, of which we in our days practice utilizes about one-twentieth. 
The following table will show how important it is to fully consume the 
combustibles to acid. One pound of carbon consumed to oxide will gen- 
erate only 1-72 horses, instead of 5-66 when consumed to acid. 

Properties of Combustion, per Hour. 



c 


CO 


CO, 





a 


A 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


cub. ft. 


1 




3-666 


2-666 


12 


149 


1 


2-666 




1-333 


6 


74-50 


0-433 


1 




0-566 


2-550 


31-65 


0-272 




1 


0-727 


3-275 


40-56 




.1-750 


1-375 


1 


3-500 


43-33 




0-445 


0-392 


0.222 


1 


12-38 




•0358 


•0246 


•0231 


•0808 


1 




0-584 


0-244 


0-170 


0-800 


9-920 




1 1-550 


0-651 


0-470 


1 2-120 


26-30 



h 



heat. 

14500 
4400 
1650 
3960 
5440 
1210 
97-3 
966 
2558 



w 



lbs. 

15 

4-55 
5-63? 
4-100 
5-633 
1-250 
0-100 

1 
2-645 



H 



horses. 
5-660 
1-720 
1-200 
1-545 
2-125 
0-472 
0-038 
0-378 
1 



284 



Weight of Boilers and Engines. 



To Approximate the Weight of Steam Boilers. 

The area of fire grate gives a nearer approximation to the weight of 
Marine boilers, than the heating surface. 

Letters denote. 
£=g = total fire grate in square feet. 

W — weight of the boiler in pounds, including fire bars, doors, smoke 
pipe, fire tools and appendages, but without water. PF=800 £3 . 
Example. Required the weight W=1 of a steam boiler of E3 =250 
square feet, grate surface. 

^=800X250=200,000 lbs. 
Weight of the water is about three-fourths of FT or of the total weight 
of boilers. 

Weight of rivets, braces or stays, doors and fire bars, is about one 
quarter of W or of the total weight of boilers. 

To Approximate the Weight of Engines. 

Letters denote 

W= weight of engine in pounds, including engine room tools, oil and 
tallow tanks, wheels, propeller and shafts. 

coefficient k. 
Trunk and oscillating engines, -------4 

Direct action paddle wheel engines, ------ 4-25 

Horizontal direct action propeller engine, - - - - 4*5 

Geared propeller engines, --------5* 

American overhead beam engines, ------ 6*5 

Side lever engines, ----------6* 

Horizontal direct action high pressure, ----- 3-5 

W=JcD*yS. 
Example. Require the weight W=A of a pair of Horizontal direct ac- 
tion propeller engines of D=72, <S=36 inches, &=4*5. 

W= 4-5X72V36 = 139968 lbs. for one cylinder, multiplied by 2=279936 
lbs. the weight required. 
For trunk engines must be taken the largest diameter. 



Punching and Sheering. 285 



Punching Iron Plates. 

To punch iron plates of from £ to 1 inch thick requires 24 tons per 
square inch of metal cut ; that is, the circumference of the hole multi- 
plied by the thickness of the plate is the area cut through. 

Letters denote. 
d = diameter of the punch or hole. 
D = diameter of the hole in the die. 
t = thickness of the iron plate. 
All dimensions in 16ths of an inch. 

W= weight oribrce in pounds required to punch the hole. 
W= 660 t d. D = tf-t-G"2 U 

Example 1. An iron plate of t—6 sixteenths of an inch thick, and the 
hole to be d=l2 sixteenths in diameter. Required the force W=1 
W= 660X6X12-47520 lbs., the answer. 
Example 2. Under the same conditions require the diameter D=1 of 
the die. 

D=12+0-2X6=13-2 sixteenths. 
Example 3. Required the diameter of piston for a direction action 
steam punch, for the plate and hole as in example 1, pressure of steam to 
be 50 lbs. per square inch. 

47520 

Force 47520=^X50 °f Vhich area of piston will be A= = 950*4 

50 

square inches, which answers to a diameter of 34*8, say 36 inches. 
Shearing Iron Plates. 

It requires the same force per section cut, for shearing as for punching, 
namely, 20 to 24 tons per square inch. If the shears are good, sharp, and 
well adjusted, 16 tons may be sufficient. 

When the cutters in the shears are inclined to one another, the area 
cut, will be the square of the thickness of the plate multiplied by half 
the cotagent for the angle of the shears. Let v=angle of the shears, W 
and t same as for punching. 

TT=88 P cot.v. 

Example 4. What force is required to cut a half inch plate t =8 sixteenths 
with a pair of shears forming an angle of v=--L2°1 Cot.l2°=4*7, 
JF=88X8 2 X 4-7=26470 lbs. 

Atmospheric Columns. 

Water=33-95 feet. 2-3 feet for 1 lbs. per square inch. 
Seawater=33-05 ft. 2-23 " " 

Mercury at 60°=30 inches. 2-05 inches, " 

Atm. air=28183 feet. 1912 feet, " " 

Atmospheric air Required for each. 

Blacksmith's forge, - - 100 to 200 
Charcoal forge, - - - - 400 to 500 

Finery forge, - - - - 800 to 1000 )- Cubic feet per minute. 
Charcoal furnace, - - 1000 to 3000 
Anthracite furnace - - 2000 to 5000 . 
Cupola. 
In a cupola of 3 feet 4 inches diameter, and 10 feet high, can be melted 
down 1000 lbs. of cast iron, 200 lbs. of bitumninous coal per hour, with a 
blowing machine of 4-5 horses making 1700 cubic feet of air per minute 
into a pressure of 2-25 inches of mercury at which the temperature of 
the blast will be about 70° Fah. 



286 Blowing Machines. 



BLOWING MACHINES. 

Letters denote 

^r^Wtafei^J^MowlnRcylider double acting. 
I = part of the stroke S under which the air compresses from the 
atmospheric density to that in the reservoir. 

F= mean resistance in pounds per square inch of the air on the 
cylinder piston. 

P = pressure in pounds per square inch of the blast in-the reservoir. 

C= cubic feet of air of atmospheric density, delivers from the blow- 
ing cylinder to the reservoir per minute. 

H= actual horse power required to work the blowing engine, includ- 
ing 13 per cent, for friction. 

d = diameter of blast pipe in inches. 

n = number of revolutions or double stroke per minute. 

A = area of supply valve to the blowing cylinder in square inches, at 
each end of cylinder. 

p = vacuum in pounds per square inch, on the supply side of the cylinder 
piston, which should not exceed 0-1 lbs. 

V= velocity of the blast through the tuyeres in feet per second. 

v = velocity of the air through the supply vaj,ve A, in feet per second, 
which should not exceed 100 feet. 

a = area of the orifice or tuyeres in square inches. 

h = height of mercury in inches, in the gauge on the blast reservoir. 

L = length of the blast pipe in feet from the receiver to the tuyeres. 

k = volume coefficient, see Table. 

t=- temperature Fah. of the blast caused by compression or heating. 

Example 1. Formulce 8. For an Anthracite blast furnace is required 
4000 cubic feet of air per minute, under a pressure of 6 inches mercury. 
Required the horse power necessary for the blowing machine 1 The ef- 
fectual resistance F=2-365 lbs. see Table. Assume the vacuum to be 
i>=0-09 lbs. 

„ r . „ 4000 (2-365+0-09) M , _ 

We have H== ■ =49-6 horses. 

198 

Example 2. Formulce 10. Suppose the blast cylinder to be D=144 inches 
diameter with S=15 feet stroke. Required the number of double strokes 
per minute n=1 

96Y4000 ««.>.. 

n= = 12*3 the answer. 

144 2 X15 

Example 3. Formulce 9. Under the above conditions, require the area 
of the supply valves A=1 when the velocity v=105 feet, per second. 

. 144 2 X15X12'3 M . -■ . _ 

A— — — ^ = 911 square inches. 

40X105 

Capacity of Blast Reservoir should not be less than the following 
proportions, but more is better. 

For one single acting cylinder, 20"] 

For one double acting cylinder, 10 S- times the capacity of one cylind'r. 

Two double act. cyl. cranks at 90° 5 ) 

One double acting cylinder, same as two single acting. The more 
cylinders the less capacity required for blast reservoir. 



F= 246 \/h .1+0-00208 1) , 

P= 14-7 (£ — 1), 

f-32 
33-55' 



£ = 32 + 493 (fc—1), 
t = 33-55 P+ 32, 



Blowing Machines. 



SS7 



Formulas for Blowing Machines. 



/=- 



Sk 



"30+A ' 
P=0-49A, 
D* S n 



- 1 

- 2 



C=l:83aA(30+AJ),6 J _ V /C+10Z 



c= 



(7= 



96 ' 
198 # 



3 H= 



19000 
C (F+p) 



u=350 y~p - 12 



198 



4A ' 



- 13 



F+p' 



\ D*Sn 

~~ 40u ' " i ^ 180000000 4 ! 



ah V 
C= , - - 5 

26 ' 



-,,14 



96 C , < 30+A 



Table for Blast and Blowing Machines. 



Volume and temperat, 



1-002 


33° 


1-005 


34*5 


1-007 


35*5 


1-010 


37 


1-012 


38 


1-015 


39*5 


1-020 


42 


1-025 


44*5 


1-030 


47 


1-035 


49-5 


1-043 


53-5 


1-051 


57*5 


1-062 


63 


1-074 


69 


1-082 


73 


1-091 


77.3 


1-100 


82 


1-109 


86-5 


1-116 


90 


1-132 


98 


1-165 


114-5 


1-200 


132 


1-265 


164-5 


1-400 


232 


1-500 


2S2 


1-625 


344-5 


1750 


407 


1-875 . 


469-5 


2-000 | 


532 



1 

2 

3 

4 
5 

6 
8 

10 

12 

14 

17 

20 

24 

28 

31 

34 

37 

46 

47-5 

54-3 

67-7 

81-4 
108-5 
163 
203-7 
254-6 
305-5 
356-4 
407-4 



0-073 
0-147 
0-220 
0-294 
0-368 
0-441 
0-588 
0-736 
0-884 
1-030 
1-250 
1-470 
1-766 
2-060 
2-281 
2-501 
2-720 
3-000 
3-500 
4-000 
5-000 
6-000 
8-000 
12-00 
15-00 
18-75 
22-49 
26-24 
30-00 



Pressure lbs. sq inch.. 
P F 



0-036 
0-079 
0-108 
0-144 
0-180 
0-216 
0-288 
0-360 
0-432 
0-503 
0-612 
0-719 
0-863 
1-008 
1-116 
1-223 
1-332 
1-470 
1-715 
1-961 
2-450 
2-941 
3-925 
5-900 
7-375 
9-217 
11-06 
13-90 
14-75 



0-032 
0-063 
0-095 
0-128 
0-159 
0-191 
0-253 
0-309 
0-379 
0-437 
0-531 
0-623 
0-745 
0-865 
0-955 
1-043 
1-130 
1-205 
1-431 
1-636 
2-010 
2-365 
3-OSS 
4-389 
6-875 
8-831 
10-67 
11-64 
12-50 



Stroke. 
I 



0-0024 

0-0049 

0-0073 

0-0097 

0-0121 

0-0145 

0-0192 

0-0239 

0-0287 

0-0334 

0-0400 

0-0467 

0-0556 

0-0643 

0-0706 

0-0769 

0-0833 

0-0908 

0-1045 

0-1178 

0-1431 

0-1667 

0-2105 

0-2S59 

0-3333 

0-3846 

0-4285 

0-4666 

0-5000 



Vetocifcv. 
V * 



72 
102 
125 
144 
161 
176 
204 
228 
249 
269 
297 
322 
352 
381 
401 
420 
438 
460 
496 
530 
593 
650 
751 
918 
1077 
1393 
1590 
1760 
1955 



2S8 



Fans or Ventilator. 



FAN OR VENTILATOR. 

Fans constructed as the accompanying 
figure have been found by the Author who 
has made several of them, to be the most 
effective. 

The vanes are each one quarter of an 
arithmetical spiral with a pitch twice the 
diameter of the fan, that is, each vane should 
be constructed in an angle of 90° from centre 
to tip. Length of fan toT)e from £ to J the 
diameter. Inlet to be half the diameter of 
the fan. Number of vanes to be not more 
than six, and not less than four. Six vanes 
work softer and better, but they give no 
better effect than four. 

The housing should be an arithmetical 
spiral with sufficient clearing for the fan at a, and leaving a space at b 
about £ of the diameter. Fans of this construction make no noise. 
Letters denote. 

1:^f r }offanin inches. 

f= l?ngtn n inVet'} of blast pipe > to be as strai S h t as possible. 

a = area in sq. in. tuyeres or outlet. 

C= cubic feet of air delivered per minute. 

h = inches of mercury. 

v = velocity in feet per second through a. 

k = volume coefficient, see Table, page 287. 

n = revolutions of fan per minute. 

H= actual horse power required to drive the fan. 




W 



Formulas for Fans. 



_ dn * 
= 60000000 



V 



11 



25 ar\-dl 



H= 



A I Tin 
24000 ' 



h = 



24000 H 
din ' 



v = nyj < */ 
28-86 \/ 



24000 H 



dl 



25 a-\-d I 



dlh 
vak 



C= — 
26 ' 

C=94=ah\7h 



94 fc 



s/ 



8 
9 
10 



V = 244y h ----- i 

Example 1. A fan of d=36 inches diameter, 1=12 inches, making n=725 
revolutions per minute, area of tuyere being #=25 square inches. Re- 
quired the density of the blast in inches of mercury h=1 



Formulae 1. h = 



36X725 3 



J 25; 



36X12 



50000000 \J 25X25-J-36X12 



= 0-242 inches. 



Example 2. Under the same conditions require the cubic containt of air 
delivered per minute, C=1 fc=l-01 the nearest in the Table. 

Formula 9. C=94X25Xl'01) / 0-242 = 1167-7 cubic feet. Required the 
horse power H=1 

, n TT 36X12X0'242X725 nHtt ^ 
Formula 2. IJ= — -- — ~ — ^ — = 3-1G horses. 
24000 



Iron Furnaces. 



289 



BLAST OE IRON FUENACES. 

It is almost impossible to set up the many variable circumstances con- 
nected with the performance of Blast Furnaces, into a table form. The 
datas herein given are deduced to an average from the performances of a 
great many furnaces both in America and Europe. 

The accompanying Tables are so arranged that the numbers in Table 
I., multiplied by the numbers in Table II., gives the corresponding charge 
of Iron ore, lime stone, coal, and the produce of pig iron in pounds per 
24 hours, with the consumption of air in cubic feet per minute. 

Table II. contains the effective capacity of blast furnaces in cubic yards. 
• Example. It is required to construct a blowing machine for an Anthra- 
cite blast furnace of 12 feet diameter of boshes, height of stack 45 feet, to 
be worked with hot blast. Required the produce of pig iron per 24 hours, 
cubic feet of air per minute and actual horse power of the blowing engine 1 

Produce of pig iron 155 Table I.X123 Table IL=19065 lbs. or 8-5 tons per 
24 hours. 

Consumption of air 20X123=2460 cubic feet per minute. Suppose the 
blast to be blown into the furnace at a pressure of P=2-94 lbs., vacuum 
in the supply side in cylinder to be p=0'01 lbs. we shall have the required 
actual power. 2460(2-364-0-07) 

Forumkz 8, p. 287. H = -! -= 30-2 horses. 

Table I. Iron or Blast Furnaces. 



The unit being the capacity 

of the Furnace in 

cubic yards. 



c c , , , f Cold blast, 

Soft charcoal { Warm bla J 



tt j , , r Cold blast, 

Hard charcoal { Warmbl ^ 



Goke 



Bituminous 



Anthracite 



f Cold blast, 
t Warm blast, 

rGold blast, 
(Warm blast, 



< Gold blast, 287 
t Warm blast, 373 



Charge and produce per 24 hours. 



Iron 
Ore. 



535 

700 

670 

875 

268 
350 

252 

327 



Pig 
Iron. 



215 
292 

270 
365 

108 
146 

101 
136 

115 
155 



Lime 
Stone. 



198 
256 



245 
320 



128 



92 
120 



105 
137 



Coal. 



400 
350 



400 
350 



515 

397 



515 

397 



515 
597 



Air 

per 

minute. 



24 

19 

24 
19 

26 
20 

24 

19 

26 
20 



Table II. Capacity and Dimensions of Iron Furnaces. 



Diameter of 




Boshes in 


ft- 


25 


- 8 




40 


9 




50 


10 




62 


11 




75 


12 




90 


13 




105 


14 




121 


15 




140 


16 




160 


17 




280 


18 




202 



30 



44 

55 

68 

82 

98 

115 

133 

153 

174 

197 

220 



Height of stack inj 
35 40 45 



47 

60 

74 

89 

106 

125 

145 

166 

189 

213 

239 



51 

64 

79 

96 

114 

134 

155 

178 

203 

229 

257 



54 
69 
75 
103 
123 
144 
167 
191 
217 
245 
275 



50 l 55 



58 

73 

91 

110 

130 



62 

78 

96 

117 

139 



153 163 



178 


189 


204 


217 


232 


247 


262 


279 


293 


312 



60 



65 
83 

102 
123 
147 

172 
200 
230 
261 
295- 
330 



25 



290 




Chemistry. 












Sixty-two Simple Substances, with their Symbols. Equivalents 




and Specific Gravities. 




Name. 


5 

& 

s 


> 

H 


> 

2 

o 

OQ 


N8JJ16, 


o 

.0 

s 
Mo 


I 

w 


> 

O 

& 

OQ 


Aluininiuni, 


Al 


13-7 ! 2-56 


Molybedenum, 


47-9 


8.60 


Antimony, (Stib.) 


Sb 


64-6 6-70 


Nickel, 


Ni 


29-5 


8.80 


Arsenic, 


A 5 


37-75-70 


Niobium, 


Nr 






Barium, 


Ba 


68-6J4-70 


Nitrogen, 


N 


14-2 


0-972 


Bismuth, 


Bi 


71-5 


9-82 


Norium, 


No 






Boron, 


B 


11-0 


2-00 


Osmium, 


Os 


99-7 


10-00 


Bromine, 


Br 


78-4 


3-00 


Oxygen, 





8-0 


1-102 


Cadmium, 


Od 


55-8 


8-65 


Palladium, 


Pel 


53-3 


11-35 


Calcium, 


Oa 


20-5 


1-58 


Pelopium, 


Pe 






Carbon, 





6-1 


3-50 


Phosphorus, 


P 


15-9 


1-77 


Cerium, 


Oe 


46-0 




Platinum, 


Pt 


98-8 


21-50 


Chlorine, 


01 


35-5 


2.44 


Potassium, (Kal.) 


K 


39-2 


0-865 


Chromium, 


Or 


26-2 


5.90 


Rhodium, 


R 


52-2 


11-00 


Cobalt, 


Oo 


29-5 


8.53 


Ruthenium, 


Ru 


52-1 


8-60 


Oolumbium, (Ta.) 


Ta 


184-8 


6.00 


Selenium, 


Se 


40-0 


4-5 


Copper, (Oopr'm) 


On 


31-7 


8-80 


Silicon, 


Si 


22-0 




Didymium, 


D 


48-0 




Silver, (Argent'm) 


Ag 


108-3 


10-5 


Erbium, 


E 






Sodium, (Natr.) 


Na 


23-5 


0-972 


Fluorine, 


F 


18-7 


1-32 


Strontium, 


Sr 


43-8 


2-54 


Glucinum, 


<T 


6-9 




Sulphur, 


S 


16-1 


1-99 


Gold, (Aurum) 


Au 


196-6 19-30 


Tellurium, 


Te 


64-2 


6-30 


Hydrogen, 


II 


1 0-069 


Terbium, 


Tb 






Iodine, 


I 


1.26*5 4-94 


Thorium, 


Th 


60-0 




Iridium, 


Tr 


98-5 


18-68 


Tin, (Stannum) 


Sn 


58-9 


7-29 


Iron, (Ferrum) 


Fe 


28-0 


7-75 


Titanium, 


Ti 


24-5 


5-28 


Lantanium, 


Ln 


48-0 




Tungsten, (Wolf.) 


W 


92-0 


17-00 


Lead, (Plumbum) 


Pb 


103-7 


11-35 


Uranium, 


U 


60-0 


10-15 


Lithium, 


L 


7-0 


0-59 


Vanadium, 


V 


68-5 




Magnesium, 


Mg 


12-7 


1-75 


Yttrium, 


Y 


32-0 




Manganese, 


Mo 


26-0 


8-00 


Zinc, 


Zn 


32-3 


7-00 


Mercury (Hydra.) 


Hg 


200.0 


13-50 


'Zirconium, 


Zr 


"34-0 




Proportions of Compounds. 


Names. 


Carbon, 
1 c 


Hydrogen. 
H 


Oxygen. 
O 


Nitrogen. 

N 


Olive oil, by weight, 


1 


772 


133 


95 






Spermaceti oil, " 


- 


780 


118 


102 






Castor oil, " 


. 


740 


103 


157 






Linseed oil, " - 


- 


760 


113 


127 






Alcohol, " ■ 


. 


527 


129 


344 






Sugar, " - 


. 


432 


68 


500 






Atmosp. air, " 








770 




230 


" air by volume, 


- 


- . 


. . . 


210 




790 


Water, fresh, by wei£ 
u a it y 0h 


'ht, - - 


- - 


1 


8 






ime, 


- - 


2 


1 






India rubber by weigl 


it, - - 


853 


147 








Blood by wt. 


66-6 iron. 


665 


53 


no- 


i 


104 


Gunpowder, 75 Ni 


toe, 


13 ch 


ircoal, 


12 Sulp 


tiur. 















Chemistry. 




291 


Binary Compounds, with 


their Formulas and Equivalents, 


Xaxne of Compound. 


Vormu'a. 

HO 


Equiv. 

9-0 


Naifle of Compound. 


.Formula. 
Pb 2 


Equiv. 

119-7 


Water, 


Binoxide of Lead, 


Binoxide of Hydro., 


H0 2 


17.0 


Dinoxide of Copper, 


Cu 2 


71-4 


Protoxide of Nitro., 


NO 


22.2 


Protoxide of Copper, 


Cu 


39 7 


Binoxide of Nitrog., 


NO2 


3U-2 


Chloride of Copper, 


Cu CI 


67-2 


Hyponitrous Acid, 


NOs 


38-2 


Oxide of Zinc, 


ZnO 


40-3 


Nitrous Acid, 


NO4 


46-2 


Sesquioxide of Ant., 


Sb 2 O3 


153-2 


Nitric Acid, 


NOs 


54-2 


Antirnonious Acid, 


Sb2 04 


161-2 


Ammonia, 


NH 3 


17-2 


Antimonic Acid, 


Sb 2 o 5 


169-2 


Cyanogen, 


NCi 


26-4 


Protoxide of Tin, 


SnO 


66-9 


Sulphurous Acid, 


SO2 


32-1 


Binoxide of Tin, 


Sn O2 


74-9 


Sulphuric Acid, 


S0 3 


40-1 


Bisulphide of Tin, 


Sn S2 


91-1 


Carbonic Oxide, 


CO 


14-1 


Chloride of Tin, 


SnCl 


94-4 


Carbonic Acid, 


C02 


22-1 


Bichloride of Tin, 


Sn Cl 2 


129-9 


Light Carb'ett'd Hy. 


H 2 C 


8-1 


Oxide of Bismuth, 


Bi 


79-5 


Olefiant Gas, 


H 2 C2 


14-2 


Chloride of Bismuth 


Bid 


107-0 


Bisulphide of Carb., 


CS2 


38-3 


Protoxide Mangan., 


MnO 


34-0 


Boracic Acid, 


B0 3 


35-0 


Sesquioxide Manga. 


Mn 2 Os 


76-0 


Chlorous Acid, 


C104 


67*5 


Red Ox. Manganese 


Mna 0* 


110-0 


Chloric Acid, 


ClOs 


75-5 


BinoxideManganese 


Mn O2 


42-0 


Hydrochloric Acid, 


HC1 


36*5 


Protoxide of Cobalt, 


CoO 


37-5 


Quadrochloride Nit., 


NCU 


156-2 


Peroxide of Cobalt, 


C02 Os 


83-0 


Iodic Acid, 


I0 5 


166-5 


Protoxide of Nickel, 


NiO 


37o 


Hydroiodic Acid, 


HI 


127-5 


Peroxide of Nickel, 


Ni 2 Os 


S3-0 


Teriodide of Nit'gn, 


NI 3 


393-7 


Ar^enious Acid, 


As 2 3 


99-4 


Hydrofluoric Acid, 


H Fl 


19-7 


Arsenic Acid, 


As 2 5 


115-4 


Phosphorous Acid, 


P2 3 


55*4 


Arseniuretted Hydr. 


H 3 A so 


7S-4 


Phosphoric Acid, 


P2O5 


71-4 


SesquisulphideArse. 


As 2 Ss 


123-7 


Phosphoret'd Hydr., 


Hs P2 


34-4 


Protoxide Mercury, 


HgO. 


208-0 


Selenious Acid, 


Se O2 


56*0 


Peroxide of Mercury 


Hg0 2 


216-0 


Selenic Acid, 


Se Os 


64-0 


Bisulphide of Merc, 


Hg S2 


232-2 


Seleniuret'd Hydro., 


H Se 


41-0 


Chloride of Mercury 


HgCl 


235-5 


Suiphurrl'd Hydro., 


HS 


17*1 


Bichloride of Merc, 


HgCl 2 


271-8 


Protoxide of Iron, 


FeO 


36-0 


Oxide of Silver, 


AgO 


116-3 


Peroxide of Iron, 


Fe 2 Os 


80-0 


Chloride of Silver, 


AgCl 


143-8 


Dinoxide of Lead, 


PbO 


111-7 


Teroxide of Gold, 


Au 3 


220-6 


Protoxide of Lead, 


Pb 2 


215-4 


Terchloride of Gold, 


Au CU 


303-1 


Quadrotrisoxi'e Le'd Pb3 Oi 


343-1 


Bichloride Platin'm. Pt CI2 


169-8 


To Transform Chemic 


il Formulas into a Mathematical 


1 


Expression. 


Rule. Multiply together th 


e equivalent, (equiv.) and the exponent, 


(exp.) of each substance and 


the product is the proportion in the com- 


pound by weight. Divide this 


by its specific gravity gives the proportions 


by bulk'or volume. 




Example 1. The chemical fo 


rmulae for common alcohol is & He O2. 


Required its proportioned par 


ts by weight in 1000] 


equiv, exp 


proportions. 


Carbon d, =6-12X4- 


=24-48) f 527 ) 

= 6 >X21'5< 129 Vbyweighk 

=16 J I 344 J 


Hy drogen Hq, = 1X6- 


Oxygen Oi, = 8X2= 


100 


1:46-48=21-5. 1000 



29? Arithmetics. 



A NEW SYSTEM OF ARITHMETIC. 

Weights, Measures a:id Coins. 

Our present Arithmetical system is very inconveniently arranged for 
the general requirements of mankind ; it causes an international difficulty 
and discordance in the adoption of a uniform system of Weights, Mea- 
sures and Coins. 

An International Association for obtaining a uniform decimal system 
of Weights, Measures and Coins, has been in existence several years, but 
as yet, has accomplished very little. They meet with the most natural 
and reasonable objections, namely, that the Arithmetical base 10 does 
not admit of binary divisions, as required in the shop and the market. 
In practice, we want our units divided into the most natural fractions, 
namely, quarters, eighths, sixteenths, &c., &c, for which the decimal 
system is not suitable. A most common fraction 1/8 expressed by decimals 
will be 0-125 ; if this number is shown to the majority of the people there 
will be comparatively few who understand the true meaning of it ; it 
will then be necessary to explain that the unit is divided into 1000 parts, 
of which 125 is 1/8 of the whole. The people will then surely remark that 
this is a roundabout way of doing things, and that they are not willing 
to cut up their things into 1000 parts' in order to get it into 8. Even 
among the educated classes and among the best arithmeticians, there will 
be few, if any who have it clearly located on the mind that 0'125 is 1/8 of 
1000, but it is very well known to be so, by practice in calculation. There- 
fore, our present arithmetical system is a great burden on the student, 
and very frequently exceeds the limits of the power of the human mind, 
beyond which solutions are performed mechanically, like a musician who 
plays the crank organ. 

The base ten has often been complained of, and more suitable numbers 
proposed. Charles the XII., of Sweden proposed the number twelve for 
the arithmetical base ; to use his own just expression that, "it is quite 
rediculous to use ten as the base for arithmetics, it can be divided once by 
two and then stops." It is not sufficient merely to propose or say that 
8, 12, or 16 would be better as a base, but in order to make a correct im- 
pression of its utility, it is necessary to enter into details with examples 
that any one may be able to see its advantages without taxing his own 
mind. The Author laid before the above mentioned International Associa- 
tion which met at Bradford in Yorkshire, on the 10th, 11th, and 12th of 
October, 1859, a new system of Arithmetics, Weights, Measures, and 
Coins, founded uniformly throughout on the number 16, as the base. 
This would become the most simple system to the mind, and it would 
embrace all requirements of the diiferent classes of mankind. 

In that system it is proposed to add six new figures, thus, 

12345678d9&'l9SSfl0 

The new figures will appear strange at the first glance, but a little re- 
flection will soon convince one of their simplicity and utility. 

A complete description with numerous examples of this new system is 
now published by J. B. Lippincott 6c Co., Philadelphia. 

In one example it will be found that our present arithmetical system 
requires, four additions, seven multiplications, and one division, em- 
ploying in the calculation 215 figures ; while the new system requires 
only one multiplication and employs only 39 figures for the same solu- 
tion. John W. Nysthom. 

Philadelphia, January, 1862. 



Optica. 293 



OPTICS. 



Optics is that branch of philosophy which treats of the property and motion 
of light. 

Mirrors* 

Example 1. Fig. 307. Before a coucave mirror of r = 6 feet radius, is placed 
an object = 1, at d = 1*75 feet from the vertex. Required the size of the 
image 1=1 

t O r 1X6 „ . 

lma S e J = 7=^T = 6-2X1-75 = 2 4 

Example 2. Fig. 308. Before a concave mirror of r — 5*25 feet radius, is placed 
an object O = 1, at D = 4*5 feet from the vertex. Required the 6ize of the in- 
verted image 1=1 

T Or 1X5-25 , . 

iniao-e I = — — = = 1*4 

2D — r 2X4-5— 5-25 

Example 3. Fig. 309. Before a convex mirror of r = 1*8 feet radius, is placed 
an object = 1, at D = 3*15 feet from the vertex. Required the size of tho 
image 1=1, and the distance in the mirror d = 1 

ima ° e I= 2xSt8=°- 222 distaMe * =2SSS-8=°- 699 ft ' 

Example 4. Fig. 310. A parabolic mirror is h = 1*31 feet high, and d = 2*15 
feet in diameter. Required the focal distance/ = 1 from the vertex. 

focal distance/= -f- - ~^ = 2-646 inches. 
16 fi 16X1*31 

Optical Lenses. 

Example 5. Fig. 316. A double convex lens, of crotvn glass, having its radii 
R = r = 6 inches. Required its principal focal distance / = 7 
For crown glass the index of refraction is m = 1*52. See table. 

/=^tif 5,768inche * 

Microscopes 

Letters denote, 
p = magnifying power of a lens. 
Jj) = limit of distinct vision. 
3 = limit of distinct sight, which for long-sighted eyes Is about 10 or 12 

inches, and near-sighted 6 to 8 inches. For common eyes take 

a = 10 inches. « 

fc = limit distance of the object from the optical centre at distinct vision. 

Example 6. Fig. 322. Required the magnifying power of a single microscope 
with principal focal distance, / = 4*3 inches 1 

Mag. power p = JL±f = 1 -^5 =3'325 times. 
/ 4-3 



255= 



294 



Optics. 



307 




Spherical Concave Mirror. 



r = radius, and f = \r, focal distance of the 
mirror. 



2 = 



Or 



dr 



r— 2 d r—2 d 

The image disappears when d =/= i r. 



308 




1 = 



Spherical Concave Mirror. 
Or ;,_ Br 



d = : 



2 B—r 2 D— r 

When the object is beyond the focal 
distance the image will be inverted. 



309 



)• T> 




Spherical Convex Mirror. 
Or , Br 



2D+r 



d = 



2I)+r 




310 



Parabolic Concave Mirror. 
d> 



f- 



d* 
16 h 



h = 



16/ 




311 



Hyperbolic Concave Mirror. 



Heat, Light, or Sound emanating from 
the foci of a hyperbola will be reflected 
diverged, from the concave surface. 



312 




Eliptic Concave Mirror. 



Emanating rays from either of the two 
foci in an elipse, will be refracted by tho 
convex surface to the other foci. 



Optic?. 295 



Astronomical Telescopes and Opera Glasses* 

Example 7. Fig. 325. The principal focal distance / = 0*65 inches of the 
ocular or eye-lens. F = 58 inches the principal focal distance of the objective- 
lens. Required the magnifying power of the telescope I — 1 



image I- 



OF 

f 



1X58 
0-65 



= 89*23 times the object. 



The telescope is to he used at the limited distance D = 1380 feet and D = co. 
Required the proper lengths I = ? and micrometrical motion of the ocular or 
eye-lens ? when the limit of distinct sight a = 10in. F= 58 : 12 = 4'833 feet. 
/ = 0-65 : 12 = 0-05416 feet. 

7 1380X4-833 10X0-05416 4*89035 



1380—4-833 ' 10+0-05416 ~ 0*05386 
When D = 1380 feet, the length I =4*94421 feet 
AYhenD = oo, Z = 4*8333 + 0*05386 =4*88719 „ 


Les. 


[0*05702 „ 
Micrometrical motion of eye lens < 0*68424 incl 

^ i 
15 nearly. 

Table of Refractive Indices* 


Substances. 


Index, 
m. 

J2-97 
\2-50 

2-55 

2-45 

1-57 

1-52 

1-63 

1-47 


Substances. 


Index, 
m. 


Cromate of Lead 

Realgar - 
Diamond - 
Glass, flint 
Glass, crown 
Oil of Cas-sia 
Oil of Olives 


Quartz - 

Muriatic Acid 

Water - 

Ice 

Hydrogen • 

Oxygen 

Atmospheric air ■ 




1.54 
1.40 
1.33 
1.30 

1.000138 
1.000272 
1.000294 




314 



JPmwi. 



An emergent rays of light a af falling upon a 
transparent medium A (say a glass prism) will be 
transmitted through in the direction a />, and de- 
livered in the direction b b', parallel to a a' a". 
V— angle of incident, v — angle of refraction. 

Indix of refraction m =■ 



sin, v 




315 Given the direction of the emergent rays a a\ 

angles e and r to find the angles u and a;,— or 

the direction of the rays b V. 



COS. Z = 



COS. 6 /to a \ 

, cos. u=mcos.[ 180— z— r). 



x = 180 —(e+r+u). 

When e = u, the angle x is smallest. 
An eye in V will see the candle in the direc- 
tion b 1 b b". 



296 



Optic?. 




316 



Double Convex Lens. 



f= + ^ r . the principal 

(m-l)(J?+r)* focal distance. 

o *= optical centre of the lens. 




317 



Piano Convex Lens. 

/= + 



m — 1 " 

The optical centre is in the convex 
surface. 




/- + ■ 



Convex-concave Lens (Meniscus.) 
Br 
(m—l)(R—r) ' 
Draw the radii R' and r' parallel to 
one another. — Draw n o, then o is the 
optical centre. 




319 



Double Concave Lens. 
Br 



f=- 



(m— l)(R+r)" 




320 



Flano Concave Lens, 
r 



/=- 



m — 1 



The optical centre is in the concave 
surface. 




321 



Concavo-convex Lens, 

/=- 



(m-l)(jK-r) 

Draw B and r' parallel to one another. 
Draw n o } then o is the optical centre. 



Optic?. 



297 




322 



Single Microscope, 
I-.O-f-.f-i, /--«£-, D-j^ 



*-•■¥> 










323 

When the object is beyond the focal 
distance the image I will be inverted. 

I:0=f:D-f, I=M, ,-*£ 



324 Diminishing Power of a Double 
Concave Lens. 

1= °/ 



I:0-f:f+D 



,-tntp. srfo 



325 



*i 



Astronomical Telescope, 



1 = 



OF 



I:0 = F:f, 

- 2>^* + a/ 
D—F^a+f 

* -f for astronomical telescope, — for opera-glasses. 






7~' 

(ifD = =o, l-F + g f ) ' 



Opera Glass. 



Formulas are the same as for Astronomical Telescope. 




298 



GE3GRAPIIY. 



GEOGRAPHY. 




The Earth on which we lire, is a round ball or sphere, with a mean diameter 
or 7914 statute miles. The whole surface of the earth is 196800000 square miles, 
cf which only one fourth, or nearly 50,000000 square miles is land, and about 
150,000000 square miles water. 

Table of Area and Population of the whole Earth* 



Divisions of tlie Earth. 

America, 

Europe, ----.-. 

Asia, -------- 

Africa, ------- 

Oceanica, ----••■ 

~Total~ 



Area in Square 

Miles. 

14.491,000 

3,760,000 

16,313,000 

10,930,000 

4j500,000__ 

50,000,000 



Population. 

60,000:000 
272.000.000 
730,000.000 
200.000.000 

27.000,000 



1,289,000,000 




AboutgT) th of the whole population are born every year, and nearly an equal 
number die in the same time; making about one born and one dead per 
second. 

The Earth is not a perfect sphere, it is flatted at the Poles. The following are 
her true dimensions in statute miles of 5280 feet. 



Dimensions of the Earth* 



Diameter 



Difference • 
Flatted- - 



Circumference 






8809 miles at the Poles. 
911-92 " mean, or in 45° lat. 

924-911 " at the Equator. 



- 26-0302 

- 13-015 

(24802-4S6 
-2 24851-640 
(24884-22 



Poles and Equator. 
at each Pole. 

round the Poles. 
Mean, or in 45° lat. 
round the Equator. 



To Find the radius of the Earth in any given Latitude 

B = 3955-96(1+ 0*00164 cos.2Z), statute mile*. 



Geography. 2UJ 



Definitions) 

Axis of the Earth is an imaginary diameter around which the earth 
revolves. 

Poles of the Earth are the two extremities of the axis, and are called North 
and South. 

Equator of the Earth is the great circle at equo-distances from the Poles; 
it divides the Earth into the Northern and Southern Hemispheres 

Meridian is any great circle of the Earth drawn through the Poles; hence 
the Meridian runs north and south, and are at right-angles to the Equator. 

The Equator and Meridians are divided into degrees, minutes, and seconds. 

Latitude is the degrees od a Meridian counted from the Equator. 

Longitude is the degrees on the Equator or on circles parallel with the 
Equator, counted at right angles from a Meridian. 

Parallels are circles drawn through equal Latitudes ; they are parallel and 
concentric with the Equator, and at right-angles to the Meridians. 

East and West is the direction of the Equator and Parallels, or at right- 
angles to north and south. Turn your face towards the south, the east is on 
the left hand, and the west on the right. 

The time in which the earth makes one revolution, is divided into 24 hours, 

360° _„ Q , 
and = lo° per hour. 

24 * 

To Reduce Longitude into Time* 

RULE. Divide the numher of degrees, minutes, and seconds hy 15, and the 
quotient will he the time. 

Example 1. Longitude 74° 48' 15", what is it in time ? . 
47i S^W the answer. 

To Find the Difference in Time between two Places* 

RULE. Divide the difference in longitude hy 15, and the quotient is the dif- 
ference in time. 

Example 2. Required the difference in time between New York and Cin- 
cinnati f 

Longitude of Cincinnati - 84°27'TF 

" " New York - - 74 07 IK 

Difference in longitude - - 10° 20' 

=-^t — = 41 minutes 20 seconds, 

* lo 

the difference in time. When it is 12 o'clock in Cincinnati, it is 12h 41' 20" in 
New York. 

Example 3. Required the difference in time between Philadelphia and Paris ? 
Longitude of Philadelphia 75° 10' W 

'« « Paris - - 2 2Q.E 

Difference in longitude 77° 30° divided by 15 will be 
6h 10 m the difference in time. When it is 12 o'clock in Philadelphia, it is oh 10 n 
o'clock in Paris. 

Example 4. A vessel sails from New York for Liverpool, after she has been at 
§ea about one week, her difference in time Tvith New York was found to be 
2A 1 m 4L^ a , Required her longitude from New York ? 

lb(2h 7 45 ) = 31° 58' 15" from New York. 



300 



Navigation. 



NAVIGATION. 






w 



i'* 



To navigate a vessel upon the supposition, that the earth is a level plane, on 
■which the meridians are drawn north and south, parallel with each other; and 
the parallels east and west, at right-angles to the former. 

The line N ' S represents a meridian north and 
south; the line WE represents a parallel east 
and west. 

A ship in I sailing in the direction of I V, and 
having reached V, it is required to know her 
position to the point 1, which is measured by the 
line IV, and the angle Nil'', and imagined by 
the lines I a and a V 

While the vessel is running from I to V, the 
distance is measured by the log and time ; and 
the course Nil' is measured by the compass 
commonly expressed in points. 
These four quantities bear the following names. 
d — IV, dista?ice from I to V in miles. 
C = Nl V, course, or points from the meridian. 
"Jj = I a, departure or difference in longitudes, in miles. 
u = a V, difference in latitudes, in miles. 
I = latitude in degrees. 
L = difference in longitude, in degrees or time. 
Formulas for Plane Sailing. 



fo = d sin.C, - 
ft = u tan.C, - 
* = 6 cos./ L , 
t» = V d*-w a , 
u = d cos.C, - 
u = fo cot. C, - 

60L cos./ 



tan.C 
u = Vd* — 

a = . — ~, 
sm.C 



1, 
2, 
3, 
4, 

5, 



7, 



*\ 



d = 
d = 



cos. C, 

60-L cos./ 
sin. C 



C0S ' Z = 607/ " 



cos./ = 



d sin.C 
"""MIT' 



10, 

11, 

12, 

13," 

14, 



cos.Z=_ Fir , - 15, 



L = 
L = 
L = 



60cos.Z ' 

d sin. C 
60cos:/ ' ' 

u tan. C 
60cos7T 

COS. C = -y , - 

a 



sin.C 



d' 



- 16, 

- 17, 

- 18, 

- *19, 

- 20, 



tan.C =-, - - 21, 

u 

sin.C = 60Lcos '\ 22, 



tan.C = 



60Z/ cos./ 



23, 



Navigation. 301 

See Table of Formulas for Plane Sailing. 

Example 1. A vessel sails east north east (6 points,) 236 miles. Required her 
departure ft ? and difference in latitude ul 

Formula 1. ft = d sin. (7 == 236Xsin.6 points = 218 miles departure, and u = d 
cos.c = 236Xcos.6 points = 90*3 miles difference in latitude. 

Example 2. A ship sails in north latitude in a course C=ESE$E = 6f 
points, at a distance of 132 miles she made a difference in longitude of L = 3° 34'. 
\Vhat latitude is she in ? 

rr t ia j dsm.O 132Xsin.Gf n , nooo 

Itomtda 14. cos J = _- = c^+S* - 0-59832, 

or I = 53° 15' the latitude. 

In high latitudes and very long distances, the preceding formulas will not 
give such correct results as may he desired, because they are set up with the 
supposition that the earth is a level plane ; but by the aid of spherical trigo- 
nometry, we are enabled to ascertain courses and distances correctly, from and 
between any known points on the earth. 

Spherical Distances* 
For the spherical formulas, letters will denote. 

I = lower latitude, in degrees from the equator. 
V = highest latitude, " " " 

C = course, from the latitude I to V, 
C" = course, from " V to I. 

d — shortest distance between I and V in degrees of the great circle. 
L = difference in longitude between I and V, in degrees, or time, 
tan.ra = coti r cos.Z. 
n = 90^1 — m. 
— I, when I and V are on one side of the equator, 
-j- l t when I is on one side and V on the other. Then 

, sin.Z' cos.w • - 

cos.d = — — , 1, 

cos.m ' 

sm.C = 5?-f_^, - . 2, 

sin.d ' 

Example. Required the shortest distance and course from New York to 
Liverpool ? 

I = 40° 42' N. latitude 1 ~ TO Vrt , 

74o « W. longitude/ ^wlork. 
V = 53° 22' N. latitude ) T . AW . 

2° 52' W. longitude | Llver P ooL , 
L = 71° 8' difference in longitude, 
tan.wi = cot 53° 22'Xcos.71° 8' = 13° 31'. 
n = 90° — 13° 31' -— 40° 42' = 35° 47'. 

Shortest distance = 47°X60-f-58 = 2878 geographical miles. 

. „ sin.71° 8'Xcob.53° 22' .^ M . 

Sm ' C - si^ito 5 8, - 490 23' = 4f points 

I course from New York NE\E. 

^ — — — 



302 


Mariners' Compass* 








S 


North. 


South. 


Points . 


Degrees. 


sineC. 


Cos.C. 


tan.C. 


N. 


* { 


i 


2° 49 / 
5 37 
8 26 


•0491 
•0979 
•1544 


•9988 
•9952 
•9880 


•04P2 
•0983 
•1982 


N. by E. 

and 
N. by W. 


S. by E. ( 

aud 1 

S. by W, ( 


1 

It 

If 


11 15 
14 4 
16 52 
19 41 


•1936 
•2430 
•2101 

•3368 

•3827 
•4276 
•4713 
•5140 


•9811 

•9700 

•9570 

•9416 

•9239 " 

•9039 

•8820 

•8577 


•1989 
•2505 
•3032 
•3577 

•4142 
•4730 
•5343 
•5993 


N. N. E. 

and 
N. N. W. 


S. S. E. r 
and J 

s. s. w. 1 


2 

2* 
2i 
21 
3 

3ir 
3£ 
3| 


22 30 
25 19 

28 7 
30 56 

33 45 
36 44 
39 22 
42 11 


N. E. by N. 
and 

N. W. by N. 


S. E. by S. ( 

and J 

S. W. by S. ( 


•5555 
•5981 
•6343 
•6715 


•8314 
-8014 
•7731 
•7410 

•7071 
•6715 
•6345 
•5981 
•5555 "" 
•5140 
•4713 
_4276_ 

•3827 
•3368 
•2901 
•2430 

•1936 
•1544 

•o:)79 

•0491 


•6883 
•7463 
•8204 
•9062 

1000 
1103 
1218 
1348 
1-496 
1-668 
1870 
2114 

2414 

2-795 
3 295 
3-991 

5027 
6-744 
1114 
20-32 _ 

QO 


N. E. 
and 
N. W. 


s, e. c 

and •} 

s. w I 


4 

4? 


45 

47 49 
50 37 
53 26 


•7071 
•7410 
•7731 
•8014 


N. E. by E. 

and 
N. W. by W. 


S. E. by E. ( 

and -J 

S. W. by W. ( 


5 

51: 
5£ 

5J 


56 15 
59 4 
61 52 
64 41 

67 30~ 
70 19 
73 7 
75 56 


•8314 

•8577 
•8820 
•9039 


E. N. E 

and 
W. N. W. 

E. by N. 

and 
W. by N. 

East or 

.. 


E. S. E. ( 

and < 

W. S. W. ( 


6 

6* 


•9239 

. -9416 

•9570 

•9700 


E. by S, ( 

and ■{ 

W. by S. ( 


7 
7£ 

7| 


78 45 
81 34 

84 22 
87 11 


•9811 
•9880 
•9952 
•9988 


West . 


8 


90° 


1000 


0.000 

1 



Navigation. 



203 



To Find the Distances of Objects at Sea* 




Height 


Distance 


Height 


Distance 


Height 


infeet. 


in miles. 


infeet. 


in miles. 


infeet. 


0-582 


1 


11 


4-39 


30 


1 


1-31 


12 


4-58 


35 


. 2 


1-87 


13 


4-77 


40 


3 


2-29 


14 


4-95 


45 


4 


2-63 


15 


5-12 


50 


5 


2-96 


16 


5-29 


60 


6 


324 


17 


5-45 


70 


7 


3-49 


18 


5-61 


80 


8 


3-73 


19 


5-77 


90 


9 


3*96 


20 


5-92 


100 


10 


4-18 


25 


6-61 


150 



7-25 

7-83 
8-37 
8-87 
9-35 
10-25 
11-07 
11-83 
12-55 
13-23 
16-22 



Height 
infeet. 

200 
300 
400 
500 
1000 
2000 
3000 
4000 
5000 
lmile. 



Distance 
in miles. 

18-72 

22-91 

26-46 

29-58 

32-41 

59-20 

72-50 

83-7 

93-5 

96-1 



The distance being the tangent a b in statute miles, at the deration a c, in 
feet. 

Example 1. The light-house at a is 100 feet above the level of the sea. 
Required the distance a b. 

Height 100 feet = 13*23 miles. 

Example 2. The flag of a ship is seen from a in d. Required the distance a, d, 
when the flag is known to be 50 feet above the level d' of the sea ? 

Height of the light 100 = 13*23 miles a, b 

Height of the flag 50 = 9-35 «__ b, d, 

Distance to the ship = 22*58 miles a, d. 

Example 3. A steamer is seen at e, the horizon b seen in the masts is assumed 
to be 16 feet above the level e'. Required the distance to the ship ? 

Height of the light 100 = 13*23 miles a b, 
The assumed height 16 = 5*2 9 " e b , 
Distance to the ship =~7*94 miles a e t 

To Find the Distance by an Observed Angle v« 

Letters denote, 
d = distance in statute miles (a e') to the object observed. 
t = the tangent (a b) in statute miles, or distance to the horizon. 
v = the observed angle eae', of the horizon and the loadline of the object. 
r = radius of the earth. 
w = the angle bac. 



COS.W = 



V? 



d = cos.(i0 — v)Vt* — r 9 — sf cos.«(u- —v )(^+r a ) — ^ 



304 



POPULATION OF COUNTRIES AND CITIES IN THE WORLD. 



Names. 
NORTH, AMERICA 

V» S. of America 
New York - - - 
Philadelphia - - 
Baltimore - - - 
Cincinnati - - - 
New Orleans - - 
Boston - - - - 
Pittsburg - - - 
St. Louis - - - 
Chicago - - - - 
Buffalo - - - - 
Louisville - - - 
Albany - - - - 
Providence - - - 
Newark, N. J. - - 
Charleston - - - 
Washington - - 
Rochester - - - 

Troy 

Richmond « - - 

Savannah - - - 

San Francisco 

British America 

Montreal - - ■ 

Toronto - - - • 

Quebec - - - « 

Halifax - - - ■ 

St. John- - - ■ 

Cuba 

Havana - - • ■ 

Santiago - • ■ 

Matanzas - - 

Puerto Princip 

Hayti 

Portuu Prince ■ 

St. Domingo - 

Jamaica 

Kingston - - 

Mexico 

Mexico City - 

Guadalaxara - 

La Puebla - - 

San Luis Potosi 

Central America 

New Guatimala - - 

SOUTH AMERICA 

Brazil 

Rio Janeiro - - - 

Bahia 

Bolivia 

La Paz ------ 

Equador 

Quito ------ 

Plata 
Buenos AyreS 
Cordova - - - - 

Paraguay 



Year 


PopulaVn 


1861 


60,000,000 


1861 


31,429,891 


1853 


850,000 


a 


575,000 


u 


195,000 


a 


160.186 


a 


145,449 


1850 


136,880 


1858 


110,241 


u 


100,000 


a 


60,625 


a 


60,000 


it 


51,726 


« 


50,703 


a 


47,500 


a 


45.500 


1850 


42,985 


a 


40,001 


1853 


40,000 


1850 


28,785 


a 


27,570 


1853 


23,458 


a 


60,000 


a 


3.634,850 


1851 


57,715 


1855 


50,000 


1852 


42,052 


1851 


33,582 


1852 


22,745 


1853 


1,009,060 


a 


147,360 


tt 


85,242 


a 


81,397 


a 


46,532 


a 


943,000 


it 


20,000 


tt 


10,000 




377,433 




35,000 




7,853,394 




200,000 




70,000 




50,000 




40,000 




2,146,000 




50,000 




16,000,000 




6,065,000 


1853 


400,000 




120,000 




1,030*000 




20,000 




500,000 




60,000 




820,000 


1854 


85,000 


" 


13,000 


1 


1,000,000 



Karnes. 
Patagonia 
Chili 
Santiago - - - - 

Valparaiso - - - 
IVew Grenada 
Bogota - - - - - 

Pern 

Lima ------ 

Venezuela 
Caraccas - - - - 

Great Britain & 

Ireland 
London - - - - 

Manchester - - - 
Liverpool - - - - 

Glasgow 

Dublin 

Edinburg - - - - 

France 

Paris ------ 

Marseilles - - - - 

Lyons - - - - - 

Spain 

Madrid - - - - - 

Barcelona - - - - 

Portugal 
Lisbon ---«.- 
Belgium 

Brussels 

Holland 
Amsterdam - - - 
Denmark 
Copenhagen - - - 
Hamburg Free City 
Bremen ditto - - - 
Sweden & Nor- 
way 
Stockholm - - - - 

Gottenburg - - - 
Christiania - - - - 

Prussia 

Berlin - - - - - 

Austria 

Vienna - - - - - 

Italy 

Rome- - - - - - 

Naples - - - - - 

Palermo - - - - - 

Turkey 

Constantinople - - 

Russia 

St. Petersburg - - 

Moscow - - - - - 

Odessa - - - - - 

Sevastopol - - - - 

China 

Pekin - - - - - 

Canton . - - - - 



Year. 



1848 



1851 
1850 
1854 
1853 



1856 
1851 



1851 
1852 



1849 
1850 



1850 



1849 
1846 



1852 
1852 



1850 
1855 



1856 
1852 
1850 
1840 

1850 
1851 
1850 



1851 
1852 
1840 
1850 
1855 



Population 
1,200,000 
1,200,000 

80,000 

60,000 
2,363,054 

40,000 

2,279,085 

100,000 

1,419,289 

63,000 

27,686,609 

2,500,000 
401,321 
376,065 
347,001 
254,850 
158,015 
35,779,222 

1,053,262 
192,527 
156,169 
13,936,218 
260,000 
121,815 

3,471,203 
455,217 

4,359,090 
123,874 

3,962,290 
228,800 

2,412,926 

133,140 

200.690 

53,156 

4,810,812 

100,000 

30,000 

26,500 

17,178,091 

441,931 

30,514,466 

407.980 

24,733,385 

177,461 

416,475 

167,222 

35,360,000 

786,990 

60,098,821 

533,241 

349,068 

71,392 

40,000 

387,632,907 

2,000,000 

1,000,000 



Names of Places. j Latitude. Longitude 
N. AMERICA AND 
WEST INDIES. 

46° 49'iV 
44 38 „ 
43 36 „ 



Quebec, - 
Halifax, - 
Portland light 
Buffalo, 
Chicago, 

Newburyport light, 
Boston State House 
Nantucket, light, 
Newport Custom,, 
New York, 
Philadelphia, 
Cape Henlopen, 
Cincinnati, - 
St. Louis, 
Richmond, 
Washington City, 
Baltimore, 
Cape Hatteras, 
Charleston light, 
Savannah, 
Cape Florida light 
Pensacola, 
Mobile, 
New Orleans, 
Porto Rico, - 
Cape Hayti's City, 
Havana, - 
Vera Cruz, - 
Mexico, - 
Porto Bello, - 
Porto Cabello, 
Cape St. August'e 
Rio Janeiro, - 
Buenos Ayres, 
Cape Horn, - 
Valparaiso Fort, 
Panama Ft. N.E., 
San Francisco, 

ENGLAND. 

London, 
Liverpool, 
Greenwich, 
Glasgow, 
Dublin, 
Edinburgh, 
Bristol, - 
Dover, • 

FRANCE. 
Paris Observatory 
Havre de Grace, 
Cherbourgh, - 
Marseilles Observ 
Antwerp, 
Calais, - 

ITALY. 
Florence, 
Leghorn, 
Rome, St. Peter's, 
Naples, light, - 
incona, light,- 

£6* 



LATITUDE AND LONGITUDE OF PLACE3._ 

Names of Places. 

GERMANY. 
Berlin, - 
Bern, 

Rotterdam, - 
Antwerp, 
Amsterdam, - 
Bremen, 
Hague, - 
Hamburg, 
Lubeck, - 
AUSTRIA. 

Vienna, - 
Venice, - 
Trieste castle, - 
TURKEY. 
Ragusa, mole, 
Athens PhUopa., 
Salonica, 
Constantinople, 
SWEDEN AND 

NORWAY. 
Stockholm, - 
Gothenburg, - 
Chris tiauia, - 
Bergen, - 
Wis'by Gotland, 
DENMARK. 



305 



42 


53 „ 


42 


o„ 


42 


48 „ 


42 


21 „ 


41 


23 „ 


41 


29 „ 


40 


42 „ 


89 


57 „ 


38 


46 „ 


SS 


6 m 


>s 


36,, 


87 


32 „ 


3S 


53 „ 


39 


18 „ 


85 


14 „ 


82 


42 „ 


82 


5„ 


25 


41 „ 


30 


24 „ 


30 


42 „ 


29 


57 „ 


18 


29 „ 


19 


46 „ 


23 


»ti 


19 


12 „ 


19 


26 „ 


9 


34 „ 


10 


28 „ 


s 


21 S 


22 


56 S 


3* 


36 „ 


55 


59 „ 


33 


2iV 


S 


57 „ 


87 


47 „ 


51 


31" 


53 


22 " 


51 


29" 


55 


52" 


63 


23" 


55 


57 " 



£ 



7116 
63 35 
70 12 

78 55 
87 35 

70 49 

71 4 
70 3 
7119 

74 00-7 

75 10 

75 4 
84 27 

89 36 
77 27 ^ 
77 0-3 

76 37 
75 30 

79 54 

81 8 

80 5 
87 10 
87 59 

90 

66 7 
7211 

82 22 
96 9 
99 5 
79 40 
68 7 
34 57 
43 9 
58 22 

67 16 
7141 
79 31 

122 21 



Latitude. ^Longitude, 



51 27 " 
51 



48 50 » 

49 29 » 

49 38 » 
43 18 » 
51 13 " 

50 58 " 



43° 46 i 
43 32 '■ 
41 54 ■ 
40 50 ' 
43 38 



6 

2 52 

416 
6 20 
312 
2 35 . 
119 E 



2 20 
6 
137 
5 22 
4 24 
151 

1116 

10 18 

12 27 

I 1416 

i 13 30 



Copenhagen, - 
Elsineur, 

RUSSIA. 
St. Petersburg, 
Moscow, - 
Revel, - 
Riga, 

Cronstadt, 
Abo, 
Odessa, - 

SPAIN. 
Madrid, - 
Barcelona, 
Algiers light, - 
Gibralter, 
Carthagena obser. 

PORTUGAL. 

Oporto, - 
Lisbon, - 
Cape St. Vincent, 

SICILY. 
Messina, 
Palermo, 
Malta, - 

CHINA. 
Peking, - 
Canton, - 

Cape of Good nope. 
Sidney, Australia, 
Jerusalem, Pales., 



52 31 N 

46 57 " 

51 54 " 

51 13 " 

52 22 " 

53 5 " 

52 4 " 

53 33 " 
53 52 " 



48 13 « 

45 26 « 

45 39 " 

42 38 it 

37 58 « 

40 39 a 

41 1 « 



69 21 « 

57 42 « 

59 55 » 

60 24 « 
57 39 « 

55 41 « 

56 2 « 

59 56 « 

55 46 « 
59 26 tt 

56 51 « 

59 58 tt 

60 27 « 
46 27 « 

40 25 « 

41 23 « 

36 49 :t 

36 6 n 

37 36 « 



11 « 
42 " 
3 « 



13 21 13 

7 25 9 
4 28 " 
4 24 

4 51 

8 49 
416 

9 56 
10 49 



10 23 

12 21 

13 46 

18 7 ^ 

23 44 2 

22 57 a 

28 59 p 



18 4 

1157 

10 52 

5 20 

1817 



38 12 « 

38 8 " 
35 54 " 

39 54 " 
23 7 " 

34 22 S 

34 „ 

31 4S iV 



12 34 
12 37 



3019 
35 33 
24 46 
23 57 

29 51 
22 15 

30 42 



3 42 W 
211 E 
3 1" 

5 20 W 
1 1 



8 38 " 

9 9" 
9 2" 



15 35 jr 

13 22 « 

14 13 « 

116 28 " 
113 14 " 

18 30 " 
151 23 " 
37 20 " 



306 Distances by Sea. 



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510 ASTTIOXOMY. 



ASTRONOMY. 

Astronomy is that branch of Philosophy which treats of the properties of 
heavenly bodies. 
The mean Solar day is divided into 24 hours. 

* A Sidereal year ia ------- 365* 6* 9™ 9-6*. 

f An Anomalistic year is - - - - • • 365* 6* 13™ 49'S-*. 

X A Tropical year is 365* 6* 13»» 49*3*. 

* Sidereal year is the time in "which the Earth makes one revolution round 
the Sun, in reference to a fixed star. 

f Anomalistic year is the time which the Earth occupies between each 
perihelion to the sun. 

Y^Tropical year is the periodical return of seasons. 

Mean distance from the Earth to the Sun 95000000 miles or 11992 diameters 
of the Earth. 

Inclination of the Ecliptic to the Equinoctial 23° 28' 40". 

The Sun subtends an angle from the Earth of 32' 3". 

Horizontal paralax of the Sun 8*6" seconds. 

Velocity of a point at the Equator by the rotation of the Earth 152*5 feet per 
second. 

Cycle of the Sun is the period of 28 years at which the days of the week 
return to the same days at the month. 

The Moon* 

Distance from the Earth to the Moon is 237000 miles, = 30 diameters of the 
Earth, = about 025 the diameter of the Sun, or the diameter of the Sun is twice 
the diameter of the Moons orbit around the Earth. 

Diameter of the Moon is 2160 miles, or about 0*2729 of the Earth's diameter. 

Volume of the Moon compared with the Earth is 0*02024. 

Density of the Moon compared with the Earth is 0*5657. 

Mass of the Moon compared with the Earth is 0*011399. 

Inclination of the Moon's orbit to the Ecliptic 5° 8' 48". 

The Moon subtends an angle from the Earth of 31' 7". 

Mean Sidereal revolution of the Moon 27*32166 days. 

Mean Sy nodical revolution of the Moon 29*5305887 days. 

The Moon passes the meridian in periods of 24-S14 hours, or 48*-* 50* later 
every day. 

Moon's Age is the number of days from the last new-moon. 

Number of Months for the Moon's Age. 



January, 0, 


April, 


2, 


July, 5, 


October, 


8, 


February, 2, 


May, 


3, 


August. 6, 


November, 


10, 


March, 1, 


June, 


4, 


September, 8, 


December, 


10. 



To Find the Moon's Age on any given day, 
RULE. Add together the day of the month, Epact for the year, and the Tab- 
ular number for the month, the sum will be the moon's age. If it exceed 30, 
reject 30's, and the remainder will be the moon's age. 

Example. Find out if it will be moonlight at Christmas, 1855? 
Number of day = 25 in December 

Epact for 1855 (see Table) = 12 

Tabular number of December = 10 

"47™ — 30 = 17 the moon's age 
or in its third quarter, consequently moon-light at mid-night. 

Golden Number or Lunar Cycle 
is the period of 19 years at which the changes of the moon fall on the same 
days of the month. 

To find the Golden number. 
RULE. Add one to the given year, divide the sum by 19, and the remainder 
will be the Golden number. 



Almanac for the 19tu Century. 



311 







Dom. 


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Dom. 


& 






Dom. \ 


Frs. 


Dat/3. 


let- 




Yrs. 


Days. 


let- 


1 


Trs. 


Days. 


let- 






ter. 


r- 






ter. 






ter. 


1800 


Baturd.* 


FE 


4 


1834 


Saturd. 


E 


20 


1868 


Sund* 


ED 


3001 


Sunday. 


D 


15 


1835 


Sunday. 


D 


1 


1869 


Monday. 


C 


1802 


Monday. 


C 


26 


1836 


Tuesd.* 


CB 


12 


1870 


Tuesd. 


B 


1803 


Tuesday 


B 


7 


1837 


Wedsd. 


A 


23 


1871 


Wedsd. 


A 


1804 


Thursd* 


AG 


18 


1838 


Thursd. 


G 


4 


1872 


Friday*. 


GF 


1805 


Friday. 


F 


29 


1839 


Friday. 


F 


15 


1873 


Saturd. 


E 


1806 


Saturd. 


E 


11 


1840 


Sund.* 


ED 


20 


1874 


Sunday. 


D 


1807 


Sunday. 


D 


22 


1841 


Monday. 


C 


7 


1875 


Monday. 


C 


1808 


Tuesd.* 


CB 


3 


1842 


Tuesd. 


B 


IS 


1876 


Wedsd* 


BA 


1809 


Wedns. 


A 


14 


1843 


Wedsd. 


A 


2^ 


1877 


Thursd. 


G 


1S10 


Thursd. 


G 


25 


1844 


Friday.* 


GF 


11 


1878 


Friday. 


F 


1811 


Friday. 


F 


6 


1845 


Saturd. 


E 


22 


1879 


Saturd. 


E 


1812 


Sunday* 


ED 


17 


1846 


Sunday. 


D 


3 


1880 


Mond.* 


DC 


1813 


Monday. 


C 


28 


1847 


Monday. 


C 


14 


1881 


Tuesd. 


B 


1S14 


Tuesd. 


B 


9 


1848 


Wedsd*. 


BA 


25 


1882 


Wedsd. 


A 


1815 


Wedns. 


A 


20 


1849 


Thursd. 


G 


6 


1883 


Thurs. 


G 


1816 


Friday*. 


GF 


1 


1850 


Friday. 


F 


17 


1884 


Saturd* 


FE 


1817 


Saturd. 


E 


12 


1851 


Saturd. 


E 


28 


1885 


Sunday. 


D 


1818 


Sunday. 


D 


23 


1852 


Mond* 


DC 


9 


1886 


Monday. 


C 


1819 


Monday. 


C 


4 


1853 


Tuesd. 


B 


20 


1887 


Tuesd. 


B 


1820 


Weds* 


BA 


15 


1854 


Wedsd. 


A 


1 


1888 


Thurs* 


AG 


1821 


Thursd. 


G 


20 


1855 


Thursd. 


G 


12 


1889 


Friday. 


F 


1822 


Friday. 


F 


7 


1856 


Saturd.* 


FE 


20 


1890 


Saturd. 


E 


1823 


Saturd. 


E 


15 


1857 


Sunday. 


D 


dr 


1891 


Sunday. 


D 


1821 


Monda*. 


DC 


20 


1858 


Monday. 


C 


15 


1892 


Tuesd * 


CB 


1825 


Tuesd. 


B 


11 


1859 


Tuesd. 


B 


20 


1893 


Wedsd. 


A 


3826 


Wedsd. 


A 


22 


1S60 


Thurs* 


AG 


7 


1894 


Thursd. 


G 


1827 


Thursd. 


G 


3 


1861 


Friday. 


F 


IS 


1895 


Friday, 


F 


1828 


Saturd* 


FE 


14 


1862 


Saturd. 


E 


20 


1896 


Sund* 


ED 


1829 


Sunday. 


D 


25 


1863 


Sunday. 


D 


11 


1897 


Monday. 


C 


1830 


Monday. 


C 


6 


1864 


Tuesd*. 


CB 


22 


1898 


Tuesd. 


B 


1831 


Tuesd. 


B 


17 


1865 


Wedsd. 


A 


3 


1899 


Thurs. 


A 


1832 


Thurs.* 


AG 


28 


1866 


Thursd. 


G 


14 


1900 


Friday*. 


GF 


1833 


Friday. 


F 


9 


1867 


Friday. 


F 


20 









In Leap years take January,* February.* 



February, 


February* 




January 


January,* 


September 




March, 




May. 




April, 




June. 


November. 


August. 




October. 


July. 


December. 




1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


21 


22 


23 


24 


25 


26 


27 


28 


29 


30 


31 











Example. On what day i i the week will the fourth of July fall in the year 
1868? 

See Tahle 1868 = Sunday*. In the Table of months, July column, Sunday 
la on the 5th ; consequently the fourth falls on Saturday. 

To Find the Latitude of a place by the Meridian Altitude 
of the Snn< 

Letters denote. 
A = meridian altitude of the sun's centre aboye the horizon, in degrees and 
minutes. (At sea the sun's lower limb is generally observed, then for correc- 
tions of semi-diameter dip of horizon, parallex, add 12 minutes to the observed 
. altitude, and the sum will be the centre altitude very nearly.) 
j D — declination of the sun at the time of observation, to be found on 
i page 314. 

2 = latitude of the place of observation. 



312 



/ =90~ii±D, - - 1, 

A - 90 — I ± D, - ■ - - 2, 

D - + 90 ± A ±1, - -j - - 3, 

"Where the quantities have double signs, plus and minus, use the upper tne 
when the latitude and declination are of equal names ; and the lower one when 
the latitude and declination are of different names. 

Example. Cn the 25th day of October, 1S53. in north latitude was observed 
the sun's meridian centre altitude to be A = 37° 53'; the declination on that 
day was B = VHP 10'' south. Required the latitude? 

I = 90 — 37° 53' — 12° 10' = 39° 57' the latitude of Philadelphia. 
To Find the Time when the Sun Rises or Sets* 

Let v be the angle of time before or after six o'clock when the sun rises or 
sets ; this angle divided by 15 and added to, or subtracted from six o'clock will 
be the true time when the sun rises or sets. 

sin.u = tan.X) tan./, - 4, 

Example. "What time does the sun rise and set, on the 27th day of July, 
1854, in north latitude I — 42° 6'? 

c > /> v +• f D = 10° 12' in the morning. ) « .. 
Sun's decimation { ^ = 19 fi , in the eTening ° ) North. 

ein.u = tan.l9° 12'Xtan.42° 6' = 0-31454 or v = 18° 20'. 
18° 20' : 15 = 1* 13^ 20* subtract from 6h 

5*_59"»_60« 

Sun rises at 4* 46™ 40« in the morning, 
sin.v = tan.l9° 5'Xtan.42° 6' -■ 0*312611 oru = 18° 13'. 
6* 

18° 13': 15 = 1 12 56 add to 6* 

Sun sets at 1 h 12™ 5G« 
To Find the Length of Day and Night* 
RULE. Double the time when the sun sets is the length of the day. 
RULE. Double the time when the sun rises is the length of the night. 
To Find the apparent Time hy an Altitude of the Sun* 
Let L be the angle of time from 12 o'clock, (noon,) when the sun's altitude a 
is observed, ■-«.»«. ^ jj 

a = the observed altitude of the sun, (if the sun's lower limb is observed add 
12' for corrections), 
A and u, same as in the preceding examples. 

T sin.fl(l±sin.v) — • m m #5 

cos.Z/ = — v . , — ' + sin.v, - - *>, 

sm.^L 

The sign -f or — is to be used as before described. 

Example. On the 11th of May, 1853. the sun's altitude in the afternoon was 
observed to be a = 42° 30'. in the latitude I - 33° 40'; the sun's declination at 
the time of observation was D = 18° iV. Required the apparent time. 

sin.v = tan.l8°xtan.33° 40' - 0-21642, - 4, 

A - 90 — 33° 40'+18 = 74 3 20' - - 2, 

, _ sin.42^ 30' (1+0-21642) __ . 21642 _ 0-63709, 5, 
C0S 3n774°20' 

or 50° 25' is the angle L, 5 -^ - 3a 21- 4C,the apparent time of observation. 
If the altitude is taken in the forenoon, subtract the obtained time from 12ft 
and the remainder is the apparent time. 











Elements of the Plane 


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System. 








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5 



314 








Sun's Declination. 










SUN'S DECLINATION. . 


« Jan. 


Feb. 


Mar. 


Apr. 


Mat I June 


July 


Aug. i Sep. Oct. 


Nov. 


Dec. 


ro 


a 


Sou. 


Sou. 


Sou. 


Nor. 


Nor. 


Nor. 


Nor. 


Nor. Nor. 


Sou. 


Sou. 


Sou. 


£ 


ft 


o / 


o / 


o / 


o / 


o / 


o / 


o , 


o / o / 


O f 


o / 


o / 


A 


1 


22 59 


16 59 


7 26 


4 41 


15 11 


22 7 


23 6 


17 58 8 10 


3 19 


14 34 


21 53 


1 


2 


22 53 


16 42 


7 3 


5 4 


15 29 


22 15 


23 2 


17 42 7 48 


3 43 


14 53 


22 2 


2 


3 22 47 


16 24 


6 40 


5 27 


15 47 


22 22 


22 57 


17 27 i7 26 


4 


15 11 


22 11 


3 


4 ! 22 41 


16 5 


6 17 


5 50 


16 4 


22 29 


22 52 


17 117 4 


4 29 


15 80 


22 19 


4 


5|22 34 


15 48 


5 54 


6 13 


16 21 


22 36 


22 46 


16 55 6 42 


4 52 


15 48 


22 27 


5 


6|22 27 


15 29 


5 30 


6 36 


16 38 


22 42 


22 40 


16 38 6 20 


5 15 


16 6 


22 34 


6 


7 22 19 


15 11 


5 7 


6 58 


16 55 


22 48 


22 34 


16 21 5 57 


5 38 


16 24 


22 41 


7 


8 22 11 


14 52 


4 44 


7 21 


17 11 


22 53 


22 27 


16 4 5 35 


9 1 


16 41 


22 47 


8 


9|22 3 


14 32 


4 29 


7 43 


17 27 


22 59 


22 20 


15 47 5 12 


6 24 


16 59 


22 53 


9 


10 21 54 


14 13 


3 57 


8 5 


17 43 


23 3 


22 13 


15 29 '4 49 


6 47 


17 16 


22 58 


10 


11 21 44 


13 53 


3 33 


8 27 


17 58 


23 7 


22 5 


15 12 4 26 


7 10 


17 32 


23 3 


11 


12 21 35 


13 33 


3 10 


8 49 


18 13 23 11 


21 56 


14 54 4 3 


7 32 


17 48 


23 8 


12 


13 21 24 


13 13 


2 46 


9 11 


18 28 23 15 


21 48 


14 35 3 40 


7 55 


18 4 


23 12 


13 


14 21 14 


12 53 


2 22 


9 32 


18 43:23 18 


21 39 


14 17 3 17 


8 17 


18 20 


23 15 


14 


1521 3 


12 32 


1 59 


9 54 


18 57,23 20 


21 29 


13 58 2 54 


8 39 


18 36 


23 18 


15 


16 20 51 


12 11 


1 35 


10 15 


19 11 23 23 


21 20 


13 39 2 31 


9 1 


18 51 


23 21 


16 


17 20 39 


11 50 


1 11 


10 36 


19 2423 24 


21 9 


13 20 2 8 


9 23 


19 5 


23 23 


17 


18 '20 27 


11 29 48 


10 57 


19 37 23 26 


20 59 


13 11 45 


9 45 


19 20 


23 25 


18 


19|20 15 


11 8 24 


11 18 


19 50 23 27 


20 48 


12 41 1 21 


10 7 


19 34 


23 26 


19 


20 20 2 


10 46 ON 


11 38 


20 03 23 27 


20 37 


12 21 58 


10 29 


19 47 


23 27 


20 


21119 49 


10 24 23 


11 59 


20 15 23 27 


20 25 12 10 35 


10 50 


20 1 


23 27 


21 


2219 35 


10 3 47 


12 19 20 27 23 27 


20 13 11 41 11 


11 11 


20 13 


23 27 


22 


2319 21 


9 41 1 11 


12 39 20 40 23 26 


20 2 11 21012S 


11 32 


20 26 


23 27 


23 


2419 7 


9 18 1 34 


12 59 20 51 123 25 


19 48 11 36 


11 53 


20 38 


23 25 


24 


25 


18 52 


8 561 58 


13 18 21 1 23 24 


19 37 10 40.0 59 


12 14 


20 50 


23 24 


25 


26 


18 37 


8 34 2 21 


13 38 21 12 23 22 


19 2410 19 |l 22 


12 35 


21 1 


23 22 


26 


27 


18 21 


8 11 2 45 


13 57 21 22 23 20 


19 10 


9 58 1 46 


12 55 


21 12 


23 19 


27 


28 


18 6 


7 49 3 8 


14 16 21 32 S3 17 


18 56 


9 37<2 9 


13 15 


21 24 


23 16 


98 


29 


17 49 


|3 32 


14 34 21 41 23 14 


18 42 


9 15 2 33 


13 35 


21 34 


23 13 


29 


30 


17 33 


3 55 


14 53 21 50 23 10 


18 28 


8 54 2 56 


13 55 


21 44 


23 9 


80 


31 


17 16 


;4 18 


|21 59 


18 13 


8 32 1 


14 14 




23 6 


51 


Declination of the Sun and equation of time for the years 


Leap years 1852, -56, '60, *64, in New York at 6h a.m. 


1853, *57, "61, 1865, „ „ apparent noon. 


1854, -58, -62, &c. „ „ 6k p.m. 


1855, -59, -63, &c. ; , „ 12h midnight. 


Exampe 1. Required the sun's declination in New York at 10 o'clock a.m. on 


the 13th of April, 1856? 


From the table April 13 declin. 9° 11' N. 


» » » a 14 „ 9 32 


Difference 21 


Dec. 9° 12' 


Correction 21 (10—6) : 24 =» 3| add. 


Required declination 9° 15' 30" the answer. 


Note. In leap years the declination and equation of time must be taken one 


day earlier in the tables for January, and February ; for instance, declination 


on the 20th of February 1856 is 7° 4lV. 













Equation of Time. 








315 




EQUATION OF TIME. 

• 


n 


Jan t . 


Feb. I Mar. . Apr. 1 May 


June 


July i Aug.' Sep. 


Oct. 1 Nov. < Dec. 


DO 


£ Add 


Add | Add 1 Add Sub. 


Sub. 


Add ! Add 


Sub. 


Sub. i Sub. Sub. 


£ 


p 


m. s. 


m. 3. m. s. 1 m. s.'m. s. 


m. s. 


ia. s. m. s. 


m. s. 


m. s. ni. s.' m. s. 


p 


1 


4 4 


13 58 12 31 1 3 52 3 4 


2 28 


3 31 6 


13 


10 24 16 16,10 37 


l 


2 


4 33 


14 5 


12 20 


3 34 


3 U 


2 19 


3 42 5 56 


32 


10 43 16 17;10 14 


2 


3 


4 5914 11 


12 7 


3 17 


3 18 


2 9 


3 53 5 52 


51 


11 2 16 17 9 50 


3 


4 


5 27 14 17 


11 54 


2 59 


3 24 


1 59 


4 4 5 47 


1 11 


11 20 16 16 9 25 


4 


5 


5 54 14 22 


11 40 


2 41 


3 29 


1 49 


4 15 5 41 


1 30 


11 37 16 14 9 


5 


6 6 20 14 26 


11 26 


2 24 


3 34 


1 38 


4 25 5 35 


1 50 


11 55 16 11 


8 35 


6 


7 6 46 


14 29 


11 12 


2 7 


3 38 


1 27 


4 35 5 28 


2 10 


12 12 16 8 


8 9 


7 


8 7 12 


14 31 


10 57 


1 50' 3 42 


1 16 


4 44 5 20 


2 31 


12 29 16 • 4 


7 43 


8 


9, 7 37 


14 33 10 41 


1 32 3 45 


1 4 


4 53 5 12 2 51 


12 45 15 58 


7 10 


9 


10 8 1 14 33'10 26 


1 16 3 48 


53 


5 2 5 3 


3 12 


13 1 15 52 


6 48 


10 


11 8 25 14 33J10 10 


1 3 50 


41 


5 10 4 54 


3 33 


13 16 15 46 


6 21 


11 


12 8 48 14 32 9 53 


42 3 51 


29 


5 17 4 44 


3 54 


13 31 15 38 


5 53 


12 


13 9 10 14 311 9 37 


26 3 52 


16 


5 25 4 34 


4 15 


13 45,15 30 


5 24 


13 


14 9 32 14 28 ; 9 20 


11 3 53 


4 


5 31 4 23 


4 36 


13 59jl5 20 4 56 


14 


35 9 53 14 25 9 3 


OS 2 3 53 


0A9 


5 38 ; 4 11 


4 57 


14 1215 10 4 27 


15 


1610 14 14 21 8 45 


17 ; 3 52 






5 18 




16 


17 10 33 14 16 8 25 


0- 31 3 51 


34 


5 48 ' 3 46 


5 40 


14 3714 47, 3 28 


17 


1810 5214 11 8 10 


45 3 49 


47 


5 53 


3 33 


6 1 


14 48 14 34 2 58 


18 


19 11 11 14 5 7 52 


58 3 46 


1 


5 57 


3 20 


6 22 


14 59 14 21 2 28 


19 


2011 28:13 58 7 33 


1 12 3 44 


1 13 


6 1 


3 6 


6 43 


15 9 14 6 1 58 


20 


21 11 45 13 51 7 15 


1 24 3 40 


1 26 


6 4 


2 51 


7 4 


15 20 13 51 


1 28 


21 


2212 1 


13 43 6 57 


1 37 3 36 


1 39 


6 6 


2 36 


7 25 


15 29 


13 35 


58 


22 


23 12 16 


13 35 6 38 


1 48 3 31 


1 52 


6 8 


2 21 


7 45 


15 37 


13 18 


28 


23 


24 12 30 


13 25 6 20 


2 3 26 2 4 


6 10 


2 5 


8 6 


15 44 


13 


A0 4 


24 


2512 44 


13 16 6 1 


2 10 3 22 2 17 


6 11 ' 1 49 


8 26 


15 51 


12 42 


32 


25 


26112 57 


13 5 5 43 


2 21 3 15 2 30 


6 11 1 33 


8 47 


15 57 


12 22 


1 2 


26 


27 13 10 


12 55 5 24 


2 30 3 9 2 42 


6 11 1 16 


9 7 


16 2 


12 2 


1 32 


27 


2813 21 


12 43 5 6 


2 40> 3 2 2 55 


6 10 58 


9 26 


16 6 


11 42 


2 1 


28 


2913 31 




4 47 


2 48j 2 54 3 7 


6 8 41 


9 46 


16 10 


11 20 


2 31 


29 


30 13 41 




4 29 


2 56 2 46 3 19 


6 6 23 


10 5 


16 13 


10 58 


3 


30 


31 13 50 




4 11 


! 2 37, 


6 4 5 


I 


16 15 




3 28 


31 


Example 2. Required the sun's declination in San Francisco at 4h 46m p.m., 


on the 5th of Aug. 1855. 


From the table Aug. 5 declin. 16° 55' N. 


» » » » 6 „ 16 38 


Difference 17 


Long, of San Francisco 122° 


„ New York 74 


48 : 15 = 3h 12m difference in time. 


Time in S. F 4 46 


Midnight 12h — 8h = 4h X 17 : 24 = 3' nearly. 

ICO AM 


Correction add 


3 




Required declina 


tion 16° 48' tl 


le answer. 


Example 3. On the 5th day of Nov. 1857, at 2h 21m 56s apparent time p.m. 


Required the mean time ? 


Apparent time 2h 21m 56s 


Equation of time, sub. 


16 14 




Mean time 2 


h 5m 42s th 


e answer. 


U 


Note. 
le mei 


Add the equation of time to or subtract from the apparent time, is 
m time. 



310 












Moon' 


S Age 














EPACT OF THE YEAR. 


, 


d, h. ( 


I.h. 


d. h. 


d.h. 


d. h. 


d. h. 


d. h. 


d h. 


d. h. 


d. h. 


d. h. 


d. h. 


1850 : 


L851 


1852 


1853 


1854 


1855 


1856 


1857 


1858 


1859 


1860 


1861 


17 17 ! 


28 8 


910 


20 1 


1 3 


1118 


2310 


4 12 


15 3 


25 17 


7 21 


1812 


1862 1 1863 ! 


1864 1 1865 1 1866 


1867 I 1868 1 1869 1 1870 1 1871 


1872 !' 


1873 


29 3 | 10 6 J 


21 21 1 2 23 1 13 15 


24 6 1 6 8 1 1623 | 2718 | 8 20 


1915 | 


17 




EPACT OF THE MONTH. 




Jan. ] 


Feb. 


Mar. 


Apr. 1 May 


June 


July 


Aug. 


Sep. 


Oct. I Nov. 


Dec. 


] 


L 17 


4 


1 16 1 2 3 


3 14 


4 2 


513 


7 


7 11 1 8 23 


9 10 


To the time of high water, or the time when the moon passes the meridian, 


add two mim 


ites for every exceeding hour of the moon's age, for corrections. 


MOON'S POSITION. HIGH WATER IN DIFFERENT PLACES. 


5 


loon. 




Add to, or subtract from the 








High water 

in N. York. 


time of high water in N. Y. 
is the time of high water in 


Rise 
Ft. 


Q'rt'r 

"New 


Face 


Age 


South." 


d. 



h. m. 


h. m. 


the desired place. 


~5~ 


12 0p.m. 


8 37a.m. 


New York 


Oh 0m 


M 


1 


12 49 „ 


9 21 „ 


Quebec add 


8 49 


17 


00 




2 


1 38 „ 


10 2 „ 


Halifax sub. 


1 


8 


•2 





3 


2 26 „ 


10 40 „ 


Boston add 


3 54 


12 


SB 


4 


3 26 „ 


11 16 „ 


New Haven add 


2 32 


17 


3 


5 


4 ±1 


11 54 „ 


Portsmouth add 


3 4 


10 






6 


4 55 „ 


12 36p.m. 


Providence sub. 


41 




Half 


Q 


7 


6 42,, 


123 „ 


Albany add 


6 34 




IS 


8 


6 30 „ 


2 16 „ 


Amboy sub. 


39 






9 


7 19 ,, 


3 16 „ 


Sandy Hook sub. 


1 8 


6 


s 


® 


10 


8 8 „ 


4 14 „ 


Philadelphia add 


5 15 


6 


11 


8 57,, 


5 9 „ 


Cape Henlopen add 


35 


5 


«rh 




12 


9 46 „ 


6 „ 


Baltimore sub. 


4 14 


12 


H 


© 


13 


10 34 „ 


6 47 „ 


Cape Henry add 


57 


4 


Full 


14 


11 23 „ 


7 29 „ 


Washington sub. 


4 8 








15 


12 12a.m. 


8 7 „ 


Norfolk sub. 


52 


7 


K 




16 


1 1 „ 


8 43 „ 


Charleston sub. 


22 


5 


@ 


17 


1 50 „ 


9 21 „ 


Key West add 


1 16 


2 


§ 


18 


2 38 „ 


10 4 „ 


Havana add 


1 35 


3 


9 


19 


3 27 „ 


10 51 „ 


Rio Janeiro sub. 


6 35 


6 


9 




20 


4 16,, 


1142 „ 


Buenos Ayres 







*"* 




21 


5 5 „ 


12 37a.m. 


Cape Horn sub. 


3 57 


9 


Half 


15 


22 


5 54 „ 


137 „ 


Valparaiso add 


55 


6 




23 


6 42 „ 


2 39 „ 


San Francisco add 


2 23 


6 


& 




24 


7 31 „ 


3 42 „ 


Liverpool add 


2 39 


25 


p* 


o 


25 


8 20 „ 


4 43 „ 


London sub. 


6 30 


18 


g 


26 


9 9 „ 


5 41 „ 


Hull sub. 


2 37 


18 


27 


9 58 „ 


6 36 „ 


Bremen add 


2 37 




rt 




28 


10 46 „ 


7 27 „ 


Lisbon sub. 


4 37 




*1 


© 


29 


11 35 „ 


8 14 '„' 


Cape Good Hope sub. 


5 37 






29£ 


12 „ 


8 37 „ 


1 







317 



Moon's Age* HigU Water* Moon South* 



Example 1. Required the moon's age on the 25th day of September, 1855? 
Date in September 25d "| 

Epact of September 7d Vadd 

Epact of 1855 lid 18h ) 

Reject 30 13d 18h is the moon's age at noon in New York. 

Example 2. Required the time of high water in New York, and at what time 
the moon is south, on the same date as in preceding example ? 
In the annexed table we have given the moon's 

Age. South. High water. 

13 days lOh 34m 6h 47m 

Correction add 18h X2= 36m __ 36m 

Moon south at llh 10m, and 7h 23m is the time of high 

water in New York. 
Example 3. Find the time of high water in San Francisco on the 18th of 
July, 1856 . 

Date in July 18d 1 

Epact of July 3d 26h >add 

Epact of 1856 23d lOh J 

Reject 30 15d 12h the moon's age. 

In the table Age. High water. 

15 8h 7m 

Correction add 12hX2 24m 

For San Francisco add 2h 23m 



Time of high water in San Francisco lOh 54m, July 18, 1856. 
The strength and direction of the wind sometimes accelerates and sometimes 
retards the tide, in consequence the most careful and scientific calculations 
may differ anhour from the time of high water. 



SOUNDINGS, 



To Reduce Soundings to Low Water. 

Letters denote. 
T= time in hours between high and low water. 
t = time in hours from low water to the time when the soundings are 

taken. 
H= vertical rise of tide in feet from high to low water. 
h = reduction of the sounding taken at the time, t, in feet. 

v = — — 7 and h = i H (1 + cos. v), 

— cos. v when v < 90 
-f cos. v when v > 90 
Example. High water at lOh 15m p.m. 
Low water at _3h_45m „ 
Time T = 6h 30ln „ 
The sounding taken at 5h 30m „ was 16 feet 6 inches 
Time t = lh 45m 
Vertical rise H = 9*75 feet. 
Required the reduction h — 1 and tru^ sounding at low water. 

18 0X1-75 a 48 o 30 CQSt v = . 6626 2. 

Reduction h = £ X 9*75 (1— 0-66262)=l-6447 feet. 
Sounding taken at 5h 30m was 16*5 feet. 
Reduction subtract h = 1-6447 



True sounding at low water 14-8553 feet. 
„ ■ — 



318 Parabolic Construction of Ships. 



PARABOLIC CONSTRUCTION OF SHIPS. 

In this kind of construction, the load water-line and greatest immersed 
section of a vessel are parabolas with the vertex at o, Figs. 1 and 2 ; also 
the square root of the areas of immersed cross-sections, taken at any dis- 
tance fore_or abaft from j%[, are subordinates in a parabola of the formula 
y = \/pcc, where, in the conic parabola, the exponent n = 2. In the ac- 
companying tables I and II, ordinates and cross-sections are calculated 
for different exponents from 2 to 16, and the corresponding lines for the 
exponents 2, 3, 4, 6, and 10, are illustrated on plates XI and XII, from which 
any desired sharpness of a vessel can be selected. The highest exponent 
makes the fullest lines. 

Letters denote. — D = disp. in cub. ft. ; T — disp. in tons of salt water ; 
ST == greatest immersed cross-section ; ® = ordinate cross-section ; L = 
length ; I = length from 25* to stern or bow ; B = beam ; b = half the 
beam ; d — load draft of water, omitting the depth of keel ; £ = any draft 
corresponding to the displacement t ; e = depth of centre of gravity of 
displacement under water-line ; x = abscissa ; y = ordinate ; n = expo- 
nent for the parabola in the water-line, n' = exponent for jgT, and n" = 
exponent for the areas of the ordinate cross-sections ; r = index for dis- 
placement ; a = area of load water-line ; $£ = b d multiplied with the 
same coefficient as for a; k = coefficient for speed and horse-power ; M = 
nautical miles or knots per hour ; H = horse-power required for the 
speed M ; m = height of metacenter above the center of gravity of dis- 
placement ; A = area of the hull of the vessel ; a' = area of the upper 
deck, or any horizontal section of the hull at d' feet from the keel ; ]$£' 
= area of greatest cross-section from the keel to a', e' = depth of center 
of gravity of the hull from the top of d', supposing the hull to be of uni- 
form thickness ; b!? = increment of draft of water ; W = increment of dis- 
placement ; bH = increment of horse-power ; bM = increment of speed ; 
C — steamship performance — coefficient. See page 275 ; N = coefficient 
for displacement, table IV ; = tabular number in table V. All linear 
dimensions are in feet, and areas in square feet. 

Hollow lines. — Let i, Fig. 1, be the point where the hollow line is to com- 
mence ; draw through i a line parallel to the centre line ; draw the ordi- 
nate z, find the ordinate z' — 2?, make a = a', then e is the stem of the boat. 
Draw equal number of ordinates on a and a', by which the line e' i is trans- 
ferred to i e, and forms the hollow part of the water-line. Table III 
contains ordinates for hollow water-lines of z = % b. All the ordinate 
water-lines are parabolas of different orders, and the ordinate for any 
point in them can be calculated by the accompanying formulas. V= the 
whole length from jgT to e, which must be divided into 8 equal parts for 
the ordinate cross-sections of the exponent n ,f . 

Fig. 3 is a scale of displacement at different drafts of water, and for dif- 
ferent exponents of vessels. 

Fig. 4 is a diagram for laying out the ordinates in the water-line and 
cross-section. One of this should be constructed for each exponent n. 
Fig. 4 is constructed with the exponent 2, the line g h = b, Figs. 1 and 2, 
and the ordinates in the inner parabola correspond with the distances 
from g in the diagram. 

Fig. 5 represents the spring of beams ; the length is divided into 8 parts 
from each end, numbered as shown by the figure. The spring b = 1 ; the 
ordinates are calculated from the line b, exponent 2, in Table I. 

Table V is calculated from the formula 20, for the elliptic form of the 
stern rail and deck. The origin of the ellipse being at o, Fig. 8 ; the ab- 
scissa x in the centre line, and the ordinate y in the breadths. 

Table VI is calculated from formula 19, and contains the coefficient N. 

TableVll contains the coefficient 0. Suppose the size of a vessel is given 
in tons of the displacement, select the coefficient N in table VI as to the 
object and condition of the vessel, find in column N, table VII, the nearest 
number to the selected coefficient, and along that line select the coefficient 
O, which, multiplied by the cube root of the displacement, gives the re- 
quired length of the vessel ; divide this length by the number of the 
column noted at the top, and it gives the draft of water ; the beam is cal- 
culated from formula 30. 



Parabolic Construction of Ships. 



310 



r=f« r> 



$ 



y n ° 

a=^ f . . 3 

n + 1 

a A 

* = inr-a" 4 



~n' + 1 

ST 



i?d- 



, . 5 
, 6 



,= , jT . , 






Practical Formulas, 
log.b — log.cc 



log .2 — log.?/ 
" 2 n 2 " + 3 »" + 1 



11 



, • 12 



2 Vn'4-2/ xl " + 2 
m ____ (n' + l)(n + l) _ ^ 14 

[1 



log.* — log.y 



* ~~ 4 (jg; L - JD) 

_ J5 3 3 / i> 
W_ lTSWJSn/ 



® =1 



V 1 ~ pn)' 



N= 



2 »' ?< 3) 



(n'+l)(2H a/, +3n"+l) 



15 

17 

IS 
19 



20 



"='(^-J'-^") 



Formulas for Hollow Lines, 
. 31 



B=»1M, .21 



M- 



kL 



4 ffl 



, 22 



G > 
M = -^L .24 



«-^?.» 



Z = Ov/?, . 29 



/ — 



l^j 



82 



6 — 2 a 



c= Z 



o-J^ 



a = l' + c — Z, 



33 



?4 



-4 = 2 f-^QE' + <*'£)+ a', 35 

e' - ^K^ + a + ^ / n' + l \ p + 1 36 

2d / Z + a + 2 i Sr' Vn' 4-2/ N /n4-2 , 
Length of a Parabola. 

2 6 7— +1 + Jlh y p.lo g . 2 J(l + 'jE+il. • • 3' 

*J4J» 4 6 J J V >/4t» / 



320 



TABLE I. Parabolic Construction of Ships. 



Expo. 


WATER-LINE OR CROSS-SECTION. 6=1. 


a=BL><. 


Jc 


n oxn' 


1. 


2 | 3 


4 


5 


€ 


75 


7 


&T=BrfX 




2 


•2345 


•4375; 


•6094 


•7500 


•8593 


•93 


•9844 


-6666 


1-94 


2-25 


•2595 


•4766; 


•6527 


•7897 


•8899 


•9558 


•9907 


•6923 


1-98 


2-5 


•2838 


•5129 


•6912 


•8232 


•9139 


•9687 


•9944 


•7142 


2-00 


2-75 


•3073 


•5466, 


•7254 


•8513 


•9326 


•9779 


•9967 


•7333 


1-98 


3- 


•3301 


•5781 


•7558 


•8750 


•9472 


•9844 


•9980 


•7500 


1-94 


3-25 


•3521 


•6074 


•7829 


-8944 


•9587 


•9889 


•9988 


•7647 


1-91 


3-5 


•3733 


•6346! 


•8070 


•9116 


•9677 


•9922 


•9993 


•7777 


1-88 


3-75 


•3939 


•6600i 


•8284 


•9256 


•9747 


•9944 


•9995 


•7894 


1-85 


4- 


•4138 


•6836! 


•8474 


•9375 


•9802 


•9961 


•9997 


•8000 


1-82 


4-5 


•4517 


•7260 


•8794 


•9557 


•9878 


•9978 


•9998 


•8181 


1-76 


5- 


•4871 


•7627 


•9046 


•9687 


•9926 


•9990 


•9999 


•8333 


1-70 


5-5 


•5202 


•7945 


•9246 


•9779 


•9954 


• -9994 


•9999 


•8461 


1-64 


6- 


•5512 


•8220 


•9404 


•9843 


•9972 


•9997 


1-000 


•8571 


1-58 


6-5 


•5802 -8459 


•9528 


•9889 


•9983 


•9998 


1-000 


•8666 


1-52 


7- 


•6073-8665 


•9627 


•9922 


•9989 


•9998 


1-000 


•8750 


1-46 


8- 


•6564-8989 


•9767 


•9960 


•9996 


•9999 


1-000 


•8S88 


1-34 


9- 


•6993-9249 


•9854 


•9985 


•999S 


•9999 


1-000 


•9000 


1-28 


10- 


•7369-9437 


•9909 


•9990 


•9999 


1-000 


1-000 


•9090 


1-18 


12- 


•7963-9683 


•9976 


•9998 


1-00C 


1-000 


1-000 


•9231 


1-08 


16- 


•8819i-9991 


•9999 


1-000 


1-00C 


) 1-000 


1-000 


•9412 


1-00 


TABLE II. DISPLACEMENT FOR THE EXPONENT n». 


Expo. 


ORDINATE CROSS-SECTIONS © . ffi = l. 


Displacement. 


»" 


1 


2 


3 


4 


5 I 

•7384 


6 | 7 


T>=mLX- { T=$[LX 


2- 


•0545 


•1914 


•3713 


•5625 


8909 -9688 


•5333 


•0152 


2-25 


•0673 


•1795 


•4260 


•6236 


•7919 


9135 


•9815 


•5663 


•0162 


2-5 


•0805 


•2512 


•4772 


•6777 


•8352 


9383 


•9889 


•5952 


•0170 


2-75 


•0944 


•2987 


•5262 


•7247 


•8697 


9563 


•9934 


•6204 


•0177 


3- 


•1090 -3342 


•5713 


•7667 


•8972 


'9691 


•9960 


•6429 


•0184 


3-25 


•1239-3689 


•6129 


•7999 


•9191 


9779 


•9976 


•6629 


•0189 


3-5 


•1394-4027 


•6512 


•8310 


•9365 


9845 


•9986 


•6806 


•0194 


3-75 


•1551-4356 


•£862 


•8567 


•9500 


9888 


•9990 


•6968 


•0199 


4- 


•1712-4673 


•7181 


•8789 


•9608 


9922 


•9994 


•7111 


•0203 


4-5 


•2039 ! -5270 


•7736 


•9146 


•9751 


9956 


•9996 


•7364 


•0211 


5- 


•2373 


•5817 


•8183 


•9384 


•9853 


■9980 


•9998 


•7575 


•0216 


5-5 


•2706 


•6312 


•8548 


•9563 


•9908 


•9988 


•9998 


•7755 


•0221 


6- 


•3038 -6757 


•8844 


•9688 


•9944 


•9995 


•9999 


•7924 


•0227 


6-5 


•3366-7155 


•9078 


•9780 


•9966 


9996 


•9999 


•8047 


•0230 


7- 


•3688-7508 


•9268 


•9845 


•9978 


9996 


•9999 


•8170 


•0233 


8- 


•4309, -8080 


•9540 


•9920 


•9992 


9998 


1-000 


•8366 


•0239 


9- 


•4890; -8554 


•9710 


•9970 


•9996 


•9998 


1-000 


•8521 


•0244 


10- 


•5430! -8906 


•9819 


•9980 


•9998 


•9999 1-000 


•8656 


•0248 


12- 


•63411-9375 


•9934 


•9996 


•9999 


•9999 1-000 


•8861 


•0253 


16- 


-7777-9991 


•9998 


•9999 


•9999' 


1-00011-000 


•9126 


•0261 



I'hilrM 




Plate Xll 



'-Hollow Waterlines Table 111 





Cross-sections M Table 1 






}„ 



i 







( > -o s s -section s M Ta h le 111 







TABLE III. Parabolic Construction of Ships. 



321 



Exp( 



Ordinates of Hollow Water Lines. 
12 3 14 5 6 



-Kb. 

7 



Area. 
a=BLX 



V—l 



2 

2-25 

2-5 

2.75 

3 

3-2; 

3.5 

3.71 

4 

4-5 

5 

5-5 

6 

6.5 

7 

8 

9 
10 
12 
16 



•1697 
•1944 
•1960 
•2127 
•2284 
•2445 
•2596 
•2779 
•2962 
•3337 
•3744 
•4081 
•4428 
•4703 
•5081 
•5861 
•6186 
•6606 
•7415 
•8481 



3732 
4074 
4415 
4744 
5063 
5368 
5657 
5947 
6193 
6670 
7105 
7465 
7790 
8266 
8328 
8794 
9047 
9274 
9601 
9871 



•5647 
•6068 
•6460 
•6817 
•7143 
•7439 
•7706 
•7954 
•8164 
•8534 
•8839 
•9070 
•9260 
•9405 
•9533 
•9719 
•9815 
•9883 
•9954 
•9993 



•7214 
•7620 
•7443 
•8277 
•8537 
•8760 
•9895 
•9114 
•9247 
•9463 
•9619 
•9727 
•9806 
•9860 
•9902 
•9953 
•9975 
•9992 
•9997 
•9999 



8433 
8754 
9013 
9219 
9383 
9513 
9617 



9762 
9853 
9909 
9944 
9965 
9978 
9988 
9994 
9998 
9999 
1-000 
1-000 



9303 

9500 

9642 

9744 

9817 

9870 

9907 

9924 

9953 

9976 

9988 

9994 

9997J 

9998 

9999| 

1-000 

1-000 

1-000 

1-000 

1-000 



9825 
9895 
9943 
9962 
9977 
9986 
9992 
9995 
9997 
9998 
9998 
9999 



1-000 
1-000 
1-000 
1-000 
1-000 
1-000 
1-000 



•6491 
•6730 
•6939 
•7124 
•7287 
•7433 
•7654 
•7682 
•7789 
•7975 
•8139 
•8269 
•8385 
•8479 
•8582 
•8730 
•8852 
•8942 
•9110 
•9319 



0527 
0535 
0532 
0523 
0510 
0496 
0482 
0463 
0452 
0424 
0390 
0375 
0354 
0352 
0327 
0286 
0261 
■0237 
0206 
0156 



322 Parabolic Construction of Ships. 



When the exponents for the water line and cross section are selected, 
multiply half the beam with the numbers in the tables, which gives the 
corresponding distances from the centre line ; for instance, if the expo- 
nent for the water line is ti=4 first column, table III, beam b—16 feet, the 
third ordinate will be 16X0-8164=13-0624 feet. Suppose the exponent 
for the cross section to be n'^3'2^, 5=16 feet, the fifth ordinate will 
be 16X0'9587=15'3392 feet. Area of the water line a is calculated 
from formula 3, and cross section $£ from formula 5, or both are 
obtained by the coefficient in the column next to the last table. Re- 
quired the area of the water line of a vessel of L=256 feet, I?=32 feet and 
71=3-75 1 

a=256X32X°"' 7 894=6472-7648 square feet the answer. Required the 
cross section $£ of a vessel of B=21 feet drafte.d=12 feet, and the expo- 
nent 71=2-5 ] 

j£F=27X12X0-7142=:231'4 square feet the answer. 

The area of the ordinate cross sections or the immersed area of any 
frame g) is calculated from formula 18 or by the numbers in table II. 
Required the 4th cross section $ = ] in a vessel of J5=42 ft., d=l& ft., and 
the exponent ?z=4-5 and n"=5. t £g , =42X18X0*8181=618-4836 square feet, 
and $=618-48X0-9384=580-38 square feet. 

The Displacement is calculated from formula 12 or by the coefficients 
in the two last columns, table II. Required the displacement of a vessel 
of dimensions as in the preceeding example when L=325 feet? 

D=618-48X325X0-7575 = 152172 cubic feet or T=618'48X325X0'0216= 
4341-729 tons, the answer. 

When the exponents for the cross section n' and for the displacement 
n" are selected, the displacement in cubic feet is obtained by multiply- 
ing together the length L, beam B, and draft of water d, and the product 
by the number in table VI. 

Example. A vessel of T=3450 tons displacement is constructed with the 
exponents n=3, n'=6, and ?i"=3-25, drawing d=16 feet water. Required 
the draft of water when the vessel and all on board weighs 2=2160 tons? 

Formula 14. r=- ^ ^ =1*435 the indix, 
6X3-25 



r /216O _ 
-P=164 / - — -=11 
' l \/ 3450 



Formula 7. cT=164 / ^= n ' 55 feet the answer. 

. , log.3450— log.2160 - nco . 

or log cT=log.l6 = — -f =1.0624. 

1.435 

Required how much M==l the same vessel can be loadded per inch b^of 
additional draft, at a draft of cT=12 feet. 

Formula 9. i t J^X^Xl^ XtV =64 . 14 to the angwcr . 

lgl.435 

The exponent for any displacement is found by formulas 15 and 16. 
Required the centre of gravity of displacement e=1 of the vessel in the 
preceding examples, when loaded to d=16 feet. 



Formula 13. e=^(|±!) 4 /^y~ 1 =6-43 feet, the answer. 

Required the area A=1 of the immersed hull of the same vessel, 
hen a=10242 square feet. 

Formulate. A=2k /?^('618.48- r -16X325 > )- r -10242=18112 square feet, 
lswer. V 6+1 V 



Plate MV 




Paracolic Construction of Ships. 



323 



For light, draft and speed, the exponents should be selected towards the 
corner -490, Table IV ; and for freight and light draft towards the corner 
•797. For heavy freight and light draft, the proportions of the vessel 
should be selected towards the corner 36'0, Table V ; sailing yachts for 
deep water towards the corner 14'9, and ordinary vessels for deep watep 
in the middle of column 12. 

The exponent for the displacement n" generally varies very little from 
that of the water-line n ; that when selecting a desired sharpness from 
fig. 1 we can consider n' 1 == n. The proper relation between n and n" de- 
pends on taste and judgment of the ship-builder. 

CONSTRUCTION OF A PROPELLER STEAMER. 

Figs. 6, 7, and 8, Flate XIV, are constructions of a steam propeller 
of the following dimensions : L= 150 ft., B = 30 ft., d = 15 ft. The scale 
is 1-32 of an inch to the foot. The exponents are selected as follows: 
?i = 2 of the forward water-line, with hollow lines table III ; for the 
ait water-line, n = 2%, full lines table I ; $£ is located 1-16 of the length 
abaft of the middle of L, making 84'375 ft. forward, and 65625 ft. abaft of 
j%)\ Exponent for the cross-section is n' = 3, when the area will be Jff -= 
337-5 sq. ft. Exponent for the displacement n' / ==2, when, from Formula 
12, D = 26998-3 cub. ft., and T=7138 tons. From Formula 32, Z = 79-9 ft., 
which makes the parabola cut the water-line at 84*375 — 79-9 = 4-475 feet 
from the stem. Exponent for the forward parabolic rail is 7i = 4, and for 
the elliptic rail abaft n = 3. The sheer is 3 ft. forward, 1% ft. abaft, with 
45° curve. The following table contains the principal dimensions for the 
form of the vessel, which table' should always be calculated before the 
construction is commenced. . • 





Cross- 


Displ'i. 


WATER-LINE. 


Ordi- 


section. 


Tab. II. 


Tab. III. Table I. 


i>ates. 


n' = 3 


7i"=2 


71 = 2 


7ir=2£ 




m 


Half ® 


Forward 


Abaft. 


1 


4-951 


9-211 


2-547 


4-257 


2 


8-671 


32-28 


5*798 


7-693 


3 


11.33 


62-63 


8*470 


10-36 


4 


13-12 


94-89 


10-82 


12-33 


5 


14-21 


124-5 


12-65 


13-71 


6 


14-76 


150-2 


13-95 


14-53 


7 


14-97 


164-4 


14-74 


14-91 


8 


15 ft. 


168-7 


15 ft. 


15 ft. 



EAIL^ | 

blel. Table V. 



SHEER. 
Tab' ~ 

n = 4t ] n = S I 45° 45° 

Forward Abaft. iFor'rd Abaft. 



5-591 
9*230 
11-44 
12-66 
13-24 
13-44 
13'49 
13-5 ft. 



9-34 
11-24 
12-18 
12-92 
13-26 
13-42 
13-49 
13-5 ft 



0-047 
0*161 
0-366 
0-645 
1 055 
1-570 
2-195 
3 ft. 



0-C23 

0-083 
0-183 
0*322 
0-527 
0-785 
1-097 
1-5 ft 



1 -r- =5-04 feet under thewater- 



The centre of gravity of displacement will be found by Formula 13, when 

' 2 U-f-2. 
line, and *" "'" — - = 4-6875 feet forward of ,£g\ 
Her launching weight t = 310 tons. 



the medium of n=2| 
34-375 — 75 



eiucm win l/c iu 

- 1 ) f 2 ±L = 

-2/N2X + 2 



Form 14. r=<±W2g§±I> =2 .i6. 

3X2 



Required, her launching draft, <T? 

Form. 7. <f=15 P*^ == 9*716 feet. 

For further explanations, with examples, seeNystrom's Treatise on Tara- 
bolic Shipbuilding and Marine Engineering Subjects, where the Parabolic 
Construction can be acquired, and the tables and formulas in this book 
will serve as a memorandum. J. B. Lippincott & Co., Philadelphia— 
Trubner, London. 



324 



Parabolic Construction of Ships. 



TABLE Y.— For Elliptic Stem of Vessels. 



Expot. 

n 



2 

2J 

2i 
•2| 

3 

3i 

3* 

4 
Sheer 

of 
vessels. 



i 



Ordinates for ellipses of different order. 
12 3 4 5 6 7 



•3398 

•4108 

•4670 

•5174 

•5604 

•5991 

•6333 

•6906 

30° 

45° 

60° 



•4840 
•5490 
•5537 
•6514 
•6911 
•7252 
•7548 
•8021 
•0149 
•0157 
•0160 



•6616 
•7147 
•7657 
•8029 
•8331 
•8578 
•8782 
•9003 
•0582 
•0539 
•0474 



•7808 
•8274 
•8627 
•8901 
•9019 
•9275 
•9406 
•9595 
•1321 
•1221 
'1086 



•8660 
•9004 
•9252 
•9434 
•9565 
•9664 
•9740 
•9840 
•2374 
•2152 
•1972 



•9204 
•9495 
•9546 
•9749 
•9821 
•9871 
•9907 
•9950 
•3740 
•3517 
■3190 



•9682 
•9801 
•9873 
•9932 
•9948 
•9973 
•9978 
•9995 
•5449 
•5227 
•4794 



•9922 
•9958 
•9978 
•9989 
•9994 
•9996 
•9998 
•9999 
•7531 
•7313 
•6946 



TABLE YI. — To Approximate Size and Shape of Vessels. 



w 


Exponent for displacement n'\ 


d n' 


2 

•356 


2-5 


3 


3*5 


4 


5 


6 


8 
•558 


10 


&f 2 


•397 


•429 


•453 


•474 


•500 


•528 


•577 


u Ql 2.5 


•381 


•425 


•459 


•486 


•508 


•541 


•566 


•597 


•620 


-S . ( 3 


•400 


•447 


•482 


•510 


•533 


•563 


•594 


•627 


•650 


% sf 3 - 5 


•414 


•462 


•500 


•528 


•552 


•589 


•616 


•650 


•673 


<a 3^ 4 


•427 


•476 


•514 


•544 


•569 


•606 


•635 


•668 


•693 


- |1 5 


•444 


•496 


•535 


•567 


•592 


•631 


•660 


•696 


•722 


if 6 


•458 


•509 


•550 


•583 


•610 


•649 


•679 


•717 


•750 


« l\ 8 


•474 


•529 


•571 


•605 


•632 


•673 


•704 


•743 


•770 


alio 


•490 


•547 


•590 


•625 


•654 


•696 


•720 


•759 


•797 


Purpose. 


Speed and 


Freight and 


Freight and 


pas 


senge 


rs. 


pas 


senge 


rs. 


slo 


w spe 


ed. 



TABLE YII— Length of Vessels = tabular number O^/T. 



n'Sscn" 

N 



•356 
•425 

•482 
•528 
•569 
•631 
.679 
•723 
•797 

Condition. 



o^ 



Proportion of draft aad length of vessels. 
12 18 36 36 48 64 82 102 



14-9 
13-9 
13*3 
12-9 
12-5 
121 
11-8 
11-6 



19-0 

17-7 

17-0 

16-5 

16-0 

15-5 

15-1 

14-8 

11-2 143 

Yessels for 

deep water. 



23-7 
22-1 
21-2 
20-5 
20-0 
194 
18-8 
18-5 
17-9 



28-9 
26-9 
25-8 
25-0 
244 
23-5 
22-9 
22-5 
21-8 



34-0 
31-7 
30-3 
29-4 
28-7 
27-6 
26-9 
26-5 
25-6 
Ordinary 
navigation. 



38-9 
36*2 
34-7 
33-7 
30-8 
31-6 
30-8 
30-3 
29*3 



43-6 
40-6 
38-9 
37-7 
36-8 
35-4 
34-6 
34-0 
32-9 



47-2 
43-9 
42-1 
40-8 
39-8 
38-3 
37-4 
36-8 
35-6 
River steamers 
Light draft. 



48-0 
44-5 

42-7 
41*3 
40-3 
38-8 
38-0 
37-2 
36-0 



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